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Properties

Label 2-624-1.1-c5-0-16
Degree 22
Conductor 624624
Sign 11
Analytic cond. 100.079100.079
Root an. cond. 10.003910.0039
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·3-s + 76·5-s − 100·7-s + 81·9-s + 106·11-s − 169·13-s − 684·15-s + 234·17-s + 276·19-s + 900·21-s − 2.54e3·23-s + 2.65e3·25-s − 729·27-s + 8.26e3·29-s + 608·31-s − 954·33-s − 7.60e3·35-s − 2.01e3·37-s + 1.52e3·39-s + 8.84e3·41-s + 1.76e4·43-s + 6.15e3·45-s − 1.87e4·47-s − 6.80e3·49-s − 2.10e3·51-s − 2.69e4·53-s + 8.05e3·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.35·5-s − 0.771·7-s + 1/3·9-s + 0.264·11-s − 0.277·13-s − 0.784·15-s + 0.196·17-s + 0.175·19-s + 0.445·21-s − 1.00·23-s + 0.848·25-s − 0.192·27-s + 1.82·29-s + 0.113·31-s − 0.152·33-s − 1.04·35-s − 0.241·37-s + 0.160·39-s + 0.821·41-s + 1.45·43-s + 0.453·45-s − 1.23·47-s − 0.405·49-s − 0.113·51-s − 1.31·53-s + 0.359·55-s + ⋯

Functional equation

Λ(s)=(624s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(624s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 624624    =    24⋅3⋅132^{4} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 100.079100.079
Root analytic conductor: 10.003910.0039
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 624, ( :5/2), 1)(2,\ 624,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.0710685412.071068541
L(12)L(\frac12) ≈\approx 2.0710685412.071068541
L(72)L(\frac{7}{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}
ppFp(T)F_p(T)
bad2 1 1
3 1+p2T 1 + p^{2} T
13 1+p2T 1 + p^{2} T
good5 1−76T+p5T2 1 - 76 T + p^{5} T^{2}
7 1+100T+p5T2 1 + 100 T + p^{5} T^{2}
11 1−106T+p5T2 1 - 106 T + p^{5} T^{2}
17 1−234T+p5T2 1 - 234 T + p^{5} T^{2}
19 1−276T+p5T2 1 - 276 T + p^{5} T^{2}
23 1+2548T+p5T2 1 + 2548 T + p^{5} T^{2}
29 1−8266T+p5T2 1 - 8266 T + p^{5} T^{2}
31 1−608T+p5T2 1 - 608 T + p^{5} T^{2}
37 1+2010T+p5T2 1 + 2010 T + p^{5} T^{2}
41 1−8844T+p5T2 1 - 8844 T + p^{5} T^{2}
43 1−17636T+p5T2 1 - 17636 T + p^{5} T^{2}
47 1+18770T+p5T2 1 + 18770 T + p^{5} T^{2}
53 1+26970T+p5T2 1 + 26970 T + p^{5} T^{2}
59 1−41966T+p5T2 1 - 41966 T + p^{5} T^{2}
61 1−778T+p5T2 1 - 778 T + p^{5} T^{2}
67 1−12632T+p5T2 1 - 12632 T + p^{5} T^{2}
71 1+40466T+p5T2 1 + 40466 T + p^{5} T^{2}
73 1−54302T+p5T2 1 - 54302 T + p^{5} T^{2}
79 1−44656T+p5T2 1 - 44656 T + p^{5} T^{2}
83 1+69918T+p5T2 1 + 69918 T + p^{5} T^{2}
89 1+44520T+p5T2 1 + 44520 T + p^{5} T^{2}
97 1+86026T+p5T2 1 + 86026 T + p^{5} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.871667909191998464250065004736, −9.308549862898110863814994841516, −8.099945882991428801004139340040, −6.81166258214305385414927266134, −6.21464578689606795192554021999, −5.49462800225665636351914547425, −4.40562519457017763375934548505, −3.01426348550849408038430859872, −1.91501641248954210564369766840, −0.71541225848001547287852013028, 0.71541225848001547287852013028, 1.91501641248954210564369766840, 3.01426348550849408038430859872, 4.40562519457017763375934548505, 5.49462800225665636351914547425, 6.21464578689606795192554021999, 6.81166258214305385414927266134, 8.099945882991428801004139340040, 9.308549862898110863814994841516, 9.871667909191998464250065004736

Graph of the ZZ-function along the critical line