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Properties

Label 4-45e2-1.1-c19e2-0-1
Degree 44
Conductor 20252025
Sign 11
Analytic cond. 10602.310602.3
Root an. cond. 10.147210.1472
Motivic weight 1919
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 152·2-s − 8.64e5·4-s − 3.90e6·5-s − 7.36e5·7-s + 1.86e8·8-s + 5.93e8·10-s − 5.36e9·11-s + 5.17e10·13-s + 1.11e8·14-s + 4.75e11·16-s + 2.50e11·17-s − 1.11e12·19-s + 3.37e12·20-s + 8.16e11·22-s + 5.92e12·23-s + 1.14e13·25-s − 7.86e12·26-s + 6.36e11·28-s + 1.41e14·29-s + 3.75e13·31-s − 1.27e14·32-s − 3.80e13·34-s + 2.87e12·35-s − 1.36e15·37-s + 1.69e14·38-s − 7.28e14·40-s + 4.17e15·41-s + ⋯
L(s)  = 1  − 0.209·2-s − 1.64·4-s − 0.894·5-s − 0.00689·7-s + 0.491·8-s + 0.187·10-s − 0.686·11-s + 1.35·13-s + 0.00144·14-s + 1.72·16-s + 0.511·17-s − 0.791·19-s + 1.47·20-s + 0.144·22-s + 0.686·23-s + 3/5·25-s − 0.283·26-s + 0.0113·28-s + 1.80·29-s + 0.254·31-s − 0.638·32-s − 0.107·34-s + 0.00616·35-s − 1.73·37-s + 0.166·38-s − 0.439·40-s + 1.99·41-s + ⋯

Functional equation

Λ(s)=(2025s/2 ΓC(s)2 L(s)=( Λ(20−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}
Λ(s)=(2025s/2 ΓC(s+19/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 20252025    =    34⋅523^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 10602.310602.3
Root analytic conductor: 10.147210.1472
Motivic weight: 1919
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 2025, ( :19/2,19/2), 1)(4,\ 2025,\ (\ :19/2, 19/2),\ 1)

Particular Values

L(10)L(10) == 00
L(12)L(\frac12) == 00
L(212)L(\frac{21}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad3 1 1
5C1C_1 (1+p9T)2 ( 1 + p^{9} T )^{2}
good2D4D_{4} 1+19p3T+6931p7T2+19p22T3+p38T4 1 + 19 p^{3} T + 6931 p^{7} T^{2} + 19 p^{22} T^{3} + p^{38} T^{4}
7D4D_{4} 1+15024p2T+288871244163854p2T2+15024p21T3+p38T4 1 + 15024 p^{2} T + 288871244163854 p^{2} T^{2} + 15024 p^{21} T^{3} + p^{38} T^{4}
11D4D_{4} 1+5369697128T+ 1 + 5369697128 T + 12 ⁣⋯ ⁣3412\!\cdots\!34T2+5369697128p19T3+p38T4 T^{2} + 5369697128 p^{19} T^{3} + p^{38} T^{4}
13D4D_{4} 1−51722759612T+ 1 - 51722759612 T + 17 ⁣⋯ ⁣2217\!\cdots\!22pT2−51722759612p19T3+p38T4 p T^{2} - 51722759612 p^{19} T^{3} + p^{38} T^{4}
17D4D_{4} 1−14719974956pT+ 1 - 14719974956 p T + 11 ⁣⋯ ⁣3811\!\cdots\!38p2T2−14719974956p20T3+p38T4 p^{2} T^{2} - 14719974956 p^{20} T^{3} + p^{38} T^{4}
19D4D_{4} 1+1113139984504T+ 1 + 1113139984504 T + 13 ⁣⋯ ⁣3813\!\cdots\!38T2+1113139984504p19T3+p38T4 T^{2} + 1113139984504 p^{19} T^{3} + p^{38} T^{4}
23D4D_{4} 1−5929365574992T+ 1 - 5929365574992 T + 81 ⁣⋯ ⁣5481\!\cdots\!54T2−5929365574992p19T3+p38T4 T^{2} - 5929365574992 p^{19} T^{3} + p^{38} T^{4}
29D4D_{4} 1−141119811247028T+ 1 - 141119811247028 T + 12 ⁣⋯ ⁣3812\!\cdots\!38T2−141119811247028p19T3+p38T4 T^{2} - 141119811247028 p^{19} T^{3} + p^{38} T^{4}
31D4D_{4} 1−37508751850032T+ 1 - 37508751850032 T + 13 ⁣⋯ ⁣4213\!\cdots\!42T2−37508751850032p19T3+p38T4 T^{2} - 37508751850032 p^{19} T^{3} + p^{38} T^{4}
37D4D_{4} 1+1368048785415476T+ 1 + 1368048785415476 T + 16 ⁣⋯ ⁣6616\!\cdots\!66T2+1368048785415476p19T3+p38T4 T^{2} + 1368048785415476 p^{19} T^{3} + p^{38} T^{4}
41D4D_{4} 1−4174938760306988T+ 1 - 4174938760306988 T + 12 ⁣⋯ ⁣2212\!\cdots\!22T2−4174938760306988p19T3+p38T4 T^{2} - 4174938760306988 p^{19} T^{3} + p^{38} T^{4}
43D4D_{4} 1+7080534797769640T+ 1 + 7080534797769640 T + 30 ⁣⋯ ⁣9030\!\cdots\!90T2+7080534797769640p19T3+p38T4 T^{2} + 7080534797769640 p^{19} T^{3} + p^{38} T^{4}
47D4D_{4} 1−239738716958080T+ 1 - 239738716958080 T + 14 ⁣⋯ ⁣9014\!\cdots\!90T2−239738716958080p19T3+p38T4 T^{2} - 239738716958080 p^{19} T^{3} + p^{38} T^{4}
53D4D_{4} 1−29662427886344452T+ 1 - 29662427886344452 T + 12 ⁣⋯ ⁣7412\!\cdots\!74T2−29662427886344452p19T3+p38T4 T^{2} - 29662427886344452 p^{19} T^{3} + p^{38} T^{4}
59D4D_{4} 1+55456595574036584T+ 1 + 55456595574036584 T + 81 ⁣⋯ ⁣5881\!\cdots\!58T2+55456595574036584p19T3+p38T4 T^{2} + 55456595574036584 p^{19} T^{3} + p^{38} T^{4}
61D4D_{4} 1−98673648121778540T+ 1 - 98673648121778540 T + 14 ⁣⋯ ⁣7814\!\cdots\!78T2−98673648121778540p19T3+p38T4 T^{2} - 98673648121778540 p^{19} T^{3} + p^{38} T^{4}
67D4D_{4} 1−546332988026030088T+ 1 - 546332988026030088 T + 16 ⁣⋯ ⁣4216\!\cdots\!42T2−546332988026030088p19T3+p38T4 T^{2} - 546332988026030088 p^{19} T^{3} + p^{38} T^{4}
71D4D_{4} 1+385389801423355024T+ 1 + 385389801423355024 T + 32 ⁣⋯ ⁣0632\!\cdots\!06T2+385389801423355024p19T3+p38T4 T^{2} + 385389801423355024 p^{19} T^{3} + p^{38} T^{4}
73D4D_{4} 1−117641357804062868T+ 1 - 117641357804062868 T + 50 ⁣⋯ ⁣5450\!\cdots\!54T2−117641357804062868p19T3+p38T4 T^{2} - 117641357804062868 p^{19} T^{3} + p^{38} T^{4}
79D4D_{4} 1+1761854290669138800T+ 1 + 1761854290669138800 T + 30 ⁣⋯ ⁣3830\!\cdots\!38T2+1761854290669138800p19T3+p38T4 T^{2} + 1761854290669138800 p^{19} T^{3} + p^{38} T^{4}
83D4D_{4} 1−515530924759284216T+ 1 - 515530924759284216 T + 49 ⁣⋯ ⁣6249\!\cdots\!62T2−515530924759284216p19T3+p38T4 T^{2} - 515530924759284216 p^{19} T^{3} + p^{38} T^{4}
89D4D_{4} 1−5056356550608812364T+ 1 - 5056356550608812364 T + 28 ⁣⋯ ⁣3828\!\cdots\!38T2−5056356550608812364p19T3+p38T4 T^{2} - 5056356550608812364 p^{19} T^{3} + p^{38} T^{4}
97D4D_{4} 1+16093051291454986172T+ 1 + 16093051291454986172 T + 17 ⁣⋯ ⁣6217\!\cdots\!62T2+16093051291454986172p19T3+p38T4 T^{2} + 16093051291454986172 p^{19} T^{3} + p^{38} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.56662021165057240485234145258, −11.02135127925894583999189653692, −10.25046141029787806804140650818, −10.07068893461508170185778540439, −9.036468496301893248669173046555, −8.690722325514147244275830722894, −8.203806255068005084445054174470, −7.896428329086555391936976627449, −6.89055267503916909272829921405, −6.31444399817776958429544525330, −5.29768178258710883598462558533, −5.01505529164607105149681508357, −4.28160973966170914792904349431, −3.77979904861757449131981938256, −3.28175328123079863031649375441, −2.48107116695375572300846622369, −1.16516040349483337034815224045, −1.07484784366982290825541638204, 0, 0, 1.07484784366982290825541638204, 1.16516040349483337034815224045, 2.48107116695375572300846622369, 3.28175328123079863031649375441, 3.77979904861757449131981938256, 4.28160973966170914792904349431, 5.01505529164607105149681508357, 5.29768178258710883598462558533, 6.31444399817776958429544525330, 6.89055267503916909272829921405, 7.896428329086555391936976627449, 8.203806255068005084445054174470, 8.690722325514147244275830722894, 9.036468496301893248669173046555, 10.07068893461508170185778540439, 10.25046141029787806804140650818, 11.02135127925894583999189653692, 11.56662021165057240485234145258

Graph of the ZZ-function along the critical line