Properties

Label 4-45e2-1.1-c19e2-0-1
Degree $4$
Conductor $2025$
Sign $1$
Analytic cond. $10602.3$
Root an. cond. $10.1472$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

Learn more

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

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

Invariants

Degree: \(4\)
Conductor: \(2025\)    =    \(3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(10602.3\)
Root analytic conductor: \(10.1472\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 2025,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5$C_1$ \( ( 1 + p^{9} T )^{2} \)
good2$D_{4}$ \( 1 + 19 p^{3} T + 6931 p^{7} T^{2} + 19 p^{22} T^{3} + p^{38} T^{4} \)
7$D_{4}$ \( 1 + 15024 p^{2} T + 288871244163854 p^{2} T^{2} + 15024 p^{21} T^{3} + p^{38} T^{4} \)
11$D_{4}$ \( 1 + 5369697128 T + \)\(12\!\cdots\!34\)\( T^{2} + 5369697128 p^{19} T^{3} + p^{38} T^{4} \)
13$D_{4}$ \( 1 - 51722759612 T + \)\(17\!\cdots\!22\)\( p T^{2} - 51722759612 p^{19} T^{3} + p^{38} T^{4} \)
17$D_{4}$ \( 1 - 14719974956 p T + \)\(11\!\cdots\!38\)\( p^{2} T^{2} - 14719974956 p^{20} T^{3} + p^{38} T^{4} \)
19$D_{4}$ \( 1 + 1113139984504 T + \)\(13\!\cdots\!38\)\( T^{2} + 1113139984504 p^{19} T^{3} + p^{38} T^{4} \)
23$D_{4}$ \( 1 - 5929365574992 T + \)\(81\!\cdots\!54\)\( T^{2} - 5929365574992 p^{19} T^{3} + p^{38} T^{4} \)
29$D_{4}$ \( 1 - 141119811247028 T + \)\(12\!\cdots\!38\)\( T^{2} - 141119811247028 p^{19} T^{3} + p^{38} T^{4} \)
31$D_{4}$ \( 1 - 37508751850032 T + \)\(13\!\cdots\!42\)\( T^{2} - 37508751850032 p^{19} T^{3} + p^{38} T^{4} \)
37$D_{4}$ \( 1 + 1368048785415476 T + \)\(16\!\cdots\!66\)\( T^{2} + 1368048785415476 p^{19} T^{3} + p^{38} T^{4} \)
41$D_{4}$ \( 1 - 4174938760306988 T + \)\(12\!\cdots\!22\)\( T^{2} - 4174938760306988 p^{19} T^{3} + p^{38} T^{4} \)
43$D_{4}$ \( 1 + 7080534797769640 T + \)\(30\!\cdots\!90\)\( T^{2} + 7080534797769640 p^{19} T^{3} + p^{38} T^{4} \)
47$D_{4}$ \( 1 - 239738716958080 T + \)\(14\!\cdots\!90\)\( T^{2} - 239738716958080 p^{19} T^{3} + p^{38} T^{4} \)
53$D_{4}$ \( 1 - 29662427886344452 T + \)\(12\!\cdots\!74\)\( T^{2} - 29662427886344452 p^{19} T^{3} + p^{38} T^{4} \)
59$D_{4}$ \( 1 + 55456595574036584 T + \)\(81\!\cdots\!58\)\( T^{2} + 55456595574036584 p^{19} T^{3} + p^{38} T^{4} \)
61$D_{4}$ \( 1 - 98673648121778540 T + \)\(14\!\cdots\!78\)\( T^{2} - 98673648121778540 p^{19} T^{3} + p^{38} T^{4} \)
67$D_{4}$ \( 1 - 546332988026030088 T + \)\(16\!\cdots\!42\)\( T^{2} - 546332988026030088 p^{19} T^{3} + p^{38} T^{4} \)
71$D_{4}$ \( 1 + 385389801423355024 T + \)\(32\!\cdots\!06\)\( T^{2} + 385389801423355024 p^{19} T^{3} + p^{38} T^{4} \)
73$D_{4}$ \( 1 - 117641357804062868 T + \)\(50\!\cdots\!54\)\( T^{2} - 117641357804062868 p^{19} T^{3} + p^{38} T^{4} \)
79$D_{4}$ \( 1 + 1761854290669138800 T + \)\(30\!\cdots\!38\)\( T^{2} + 1761854290669138800 p^{19} T^{3} + p^{38} T^{4} \)
83$D_{4}$ \( 1 - 515530924759284216 T + \)\(49\!\cdots\!62\)\( T^{2} - 515530924759284216 p^{19} T^{3} + p^{38} T^{4} \)
89$D_{4}$ \( 1 - 5056356550608812364 T + \)\(28\!\cdots\!38\)\( T^{2} - 5056356550608812364 p^{19} T^{3} + p^{38} T^{4} \)
97$D_{4}$ \( 1 + 16093051291454986172 T + \)\(17\!\cdots\!62\)\( T^{2} + 16093051291454986172 p^{19} T^{3} + p^{38} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line