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Properties

Label 4-315e2-1.1-c3e2-0-9
Degree 44
Conductor 9922599225
Sign 11
Analytic cond. 345.424345.424
Root an. cond. 4.311104.31110
Motivic weight 33
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  + 2-s − 11·4-s + 10·5-s − 14·7-s − 15·8-s + 10·10-s − 4·11-s − 22·13-s − 14·14-s + 61·16-s − 58·17-s − 110·20-s − 4·22-s − 82·23-s + 75·25-s − 22·26-s + 154·28-s − 334·29-s − 210·31-s + 89·32-s − 58·34-s − 140·35-s + 6·37-s − 150·40-s − 176·41-s + 46·43-s + 44·44-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.37·4-s + 0.894·5-s − 0.755·7-s − 0.662·8-s + 0.316·10-s − 0.109·11-s − 0.469·13-s − 0.267·14-s + 0.953·16-s − 0.827·17-s − 1.22·20-s − 0.0387·22-s − 0.743·23-s + 3/5·25-s − 0.165·26-s + 1.03·28-s − 2.13·29-s − 1.21·31-s + 0.491·32-s − 0.292·34-s − 0.676·35-s + 0.0266·37-s − 0.592·40-s − 0.670·41-s + 0.163·43-s + 0.150·44-s + ⋯

Functional equation

Λ(s)=(99225s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(99225s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 99225 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9922599225    =    34⋅52⋅723^{4} \cdot 5^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 345.424345.424
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 99225, ( :3/2,3/2), 1)(4,\ 99225,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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−pT)2 ( 1 - p T )^{2}
7C1C_1 (1+pT)2 ( 1 + p T )^{2}
good2D4D_{4} 1−T+3p2T2−p3T3+p6T4 1 - T + 3 p^{2} T^{2} - p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+4T+2598T2+4p3T3+p6T4 1 + 4 T + 2598 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1+22T+2458T2+22p3T3+p6T4 1 + 22 T + 2458 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+58T+7794T2+58p3T3+p6T4 1 + 58 T + 7794 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4}
19C22C_2^2 1+5490T2+p6T4 1 + 5490 T^{2} + p^{6} T^{4}
23D4D_{4} 1+82T+25182T2+82p3T3+p6T4 1 + 82 T + 25182 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1+334T+72842T2+334p3T3+p6T4 1 + 334 T + 72842 T^{2} + 334 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1+210T+69230T2+210p3T3+p6T4 1 + 210 T + 69230 T^{2} + 210 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1−6T+97490T2−6p3T3+p6T4 1 - 6 T + 97490 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1+176T+134094T2+176p3T3+p6T4 1 + 176 T + 134094 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4}
43D4D_{4} 1−46T+42430T2−46p3T3+p6T4 1 - 46 T + 42430 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4}
47D4D_{4} 1+514T+269870T2+514p3T3+p6T4 1 + 514 T + 269870 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+808T+449478T2+808p3T3+p6T4 1 + 808 T + 449478 T^{2} + 808 p^{3} T^{3} + p^{6} T^{4}
59C2C_2 (1+284T+p3T2)2 ( 1 + 284 T + p^{3} T^{2} )^{2}
61D4D_{4} 1+618T+515018T2+618p3T3+p6T4 1 + 618 T + 515018 T^{2} + 618 p^{3} T^{3} + p^{6} T^{4}
67D4D_{4} 1−694T+681118T2−694p3T3+p6T4 1 - 694 T + 681118 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+814T+769934T2+814p3T3+p6T4 1 + 814 T + 769934 T^{2} + 814 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1−82T+422290T2−82p3T3+p6T4 1 - 82 T + 422290 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1−600T+618846T2−600p3T3+p6T4 1 - 600 T + 618846 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1−268T+779030T2−268p3T3+p6T4 1 - 268 T + 779030 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−72T+448286T2−72p3T3+p6T4 1 - 72 T + 448286 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1−1626T+1498938T2−1626p3T3+p6T4 1 - 1626 T + 1498938 T^{2} - 1626 p^{3} T^{3} + p^{6} 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

−10.83220766002518717995974441685, −10.57431655397137215471692263117, −9.697533450317179146007052011089, −9.662480775750861955170523573168, −9.120194146511698226614978741361, −9.018914884707597032364058327883, −8.098503659432543357784373585902, −7.78573627573184770330176879861, −6.97849430202312930360600406647, −6.52369794341279332217201960359, −5.76957120946380157339099050648, −5.64615198411599812420155516438, −4.68081441727888723927357170415, −4.65388526606627060183646159169, −3.56384232611935751687555889382, −3.37030543644790019091490755104, −2.24367500346555214169611067185, −1.60539406169862632853283362378, 0, 0, 1.60539406169862632853283362378, 2.24367500346555214169611067185, 3.37030543644790019091490755104, 3.56384232611935751687555889382, 4.65388526606627060183646159169, 4.68081441727888723927357170415, 5.64615198411599812420155516438, 5.76957120946380157339099050648, 6.52369794341279332217201960359, 6.97849430202312930360600406647, 7.78573627573184770330176879861, 8.098503659432543357784373585902, 9.018914884707597032364058327883, 9.120194146511698226614978741361, 9.662480775750861955170523573168, 9.697533450317179146007052011089, 10.57431655397137215471692263117, 10.83220766002518717995974441685

Graph of the ZZ-function along the critical line