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Properties

Label 2-338-1.1-c5-0-10
Degree 22
Conductor 338338
Sign 11
Analytic cond. 54.209754.2097
Root an. cond. 7.362727.36272
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  − 4·2-s + 4·3-s + 16·4-s − 68·5-s − 16·6-s + 82·7-s − 64·8-s − 227·9-s + 272·10-s + 390·11-s + 64·12-s − 328·14-s − 272·15-s + 256·16-s − 1.73e3·17-s + 908·18-s + 1.07e3·19-s − 1.08e3·20-s + 328·21-s − 1.56e3·22-s − 2.10e3·23-s − 256·24-s + 1.49e3·25-s − 1.88e3·27-s + 1.31e3·28-s − 1.69e3·29-s + 1.08e3·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.256·3-s + 1/2·4-s − 1.21·5-s − 0.181·6-s + 0.632·7-s − 0.353·8-s − 0.934·9-s + 0.860·10-s + 0.971·11-s + 0.128·12-s − 0.447·14-s − 0.312·15-s + 1/4·16-s − 1.45·17-s + 0.660·18-s + 0.682·19-s − 0.608·20-s + 0.162·21-s − 0.687·22-s − 0.829·23-s − 0.0907·24-s + 0.479·25-s − 0.496·27-s + 0.316·28-s − 0.373·29-s + 0.220·30-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: 11
Analytic conductor: 54.209754.2097
Root analytic conductor: 7.362727.36272
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 338, ( :5/2), 1)(2,\ 338,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 0.97547664090.9754766409
L(12)L(\frac12) ≈\approx 0.97547664090.9754766409
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+p2T 1 + p^{2} T
13 1 1
good3 1−4T+p5T2 1 - 4 T + p^{5} T^{2}
5 1+68T+p5T2 1 + 68 T + p^{5} T^{2}
7 1−82T+p5T2 1 - 82 T + p^{5} T^{2}
11 1−390T+p5T2 1 - 390 T + p^{5} T^{2}
17 1+1738T+p5T2 1 + 1738 T + p^{5} T^{2}
19 1−1074T+p5T2 1 - 1074 T + p^{5} T^{2}
23 1+2104T+p5T2 1 + 2104 T + p^{5} T^{2}
29 1+1690T+p5T2 1 + 1690 T + p^{5} T^{2}
31 1−1430T+p5T2 1 - 1430 T + p^{5} T^{2}
37 1−8852T+p5T2 1 - 8852 T + p^{5} T^{2}
41 1+6760T+p5T2 1 + 6760 T + p^{5} T^{2}
43 1−16916T+p5T2 1 - 16916 T + p^{5} T^{2}
47 1+25158T+p5T2 1 + 25158 T + p^{5} T^{2}
53 1−38214T+p5T2 1 - 38214 T + p^{5} T^{2}
59 1−21286T+p5T2 1 - 21286 T + p^{5} T^{2}
61 1+5458T+p5T2 1 + 5458 T + p^{5} T^{2}
67 1+44542T+p5T2 1 + 44542 T + p^{5} T^{2}
71 1−17790T+p5T2 1 - 17790 T + p^{5} T^{2}
73 1−31064T+p5T2 1 - 31064 T + p^{5} T^{2}
79 1+45360T+p5T2 1 + 45360 T + p^{5} T^{2}
83 1−124546T+p5T2 1 - 124546 T + p^{5} T^{2}
89 1+18744T+p5T2 1 + 18744 T + p^{5} T^{2}
97 1−121488T+p5T2 1 - 121488 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

−10.95765132372656315009809653338, −9.562017649552566375919476184828, −8.672433550967950536623230641817, −8.080597551754881335109033869998, −7.18380258388319021430183366841, −6.04965797397424007708346045658, −4.52505307535669163523688403516, −3.48827405918081510715783656158, −2.10118700321670276952613171745, −0.58144755990398679628433328036, 0.58144755990398679628433328036, 2.10118700321670276952613171745, 3.48827405918081510715783656158, 4.52505307535669163523688403516, 6.04965797397424007708346045658, 7.18380258388319021430183366841, 8.080597551754881335109033869998, 8.672433550967950536623230641817, 9.562017649552566375919476184828, 10.95765132372656315009809653338

Graph of the ZZ-function along the critical line