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Properties

Label 2-350-1.1-c3-0-14
Degree 22
Conductor 350350
Sign 11
Analytic cond. 20.650620.6506
Root an. cond. 4.544304.54430
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 10·3-s + 4·4-s − 20·6-s + 7·7-s − 8·8-s + 73·9-s + 9·11-s + 40·12-s − 52·13-s − 14·14-s + 16·16-s + 96·17-s − 146·18-s − 10·19-s + 70·21-s − 18·22-s + 75·23-s − 80·24-s + 104·26-s + 460·27-s + 28·28-s + 189·29-s − 232·31-s − 32·32-s + 90·33-s − 192·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.92·3-s + 1/2·4-s − 1.36·6-s + 0.377·7-s − 0.353·8-s + 2.70·9-s + 0.246·11-s + 0.962·12-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 1.36·17-s − 1.91·18-s − 0.120·19-s + 0.727·21-s − 0.174·22-s + 0.679·23-s − 0.680·24-s + 0.784·26-s + 3.27·27-s + 0.188·28-s + 1.21·29-s − 1.34·31-s − 0.176·32-s + 0.474·33-s − 0.968·34-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 20.650620.6506
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 350, ( :3/2), 1)(2,\ 350,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.9496267492.949626749
L(12)L(\frac12) ≈\approx 2.9496267492.949626749
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}
ppFp(T)F_p(T)
bad2 1+pT 1 + p T
5 1 1
7 1−pT 1 - p T
good3 1−10T+p3T2 1 - 10 T + p^{3} T^{2}
11 1−9T+p3T2 1 - 9 T + p^{3} T^{2}
13 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
17 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
19 1+10T+p3T2 1 + 10 T + p^{3} T^{2}
23 1−75T+p3T2 1 - 75 T + p^{3} T^{2}
29 1−189T+p3T2 1 - 189 T + p^{3} T^{2}
31 1+232T+p3T2 1 + 232 T + p^{3} T^{2}
37 1−305T+p3T2 1 - 305 T + p^{3} T^{2}
41 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
43 1−353T+p3T2 1 - 353 T + p^{3} T^{2}
47 1+486T+p3T2 1 + 486 T + p^{3} T^{2}
53 1+354T+p3T2 1 + 354 T + p^{3} T^{2}
59 1+672T+p3T2 1 + 672 T + p^{3} T^{2}
61 1−206T+p3T2 1 - 206 T + p^{3} T^{2}
67 1−599T+p3T2 1 - 599 T + p^{3} T^{2}
71 1+471T+p3T2 1 + 471 T + p^{3} T^{2}
73 1−614T+p3T2 1 - 614 T + p^{3} T^{2}
79 1−743T+p3T2 1 - 743 T + p^{3} T^{2}
83 1−12pT+p3T2 1 - 12 p T + p^{3} T^{2}
89 1−180T+p3T2 1 - 180 T + p^{3} T^{2}
97 1+184T+p3T2 1 + 184 T + p^{3} 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

−10.67578773904240973083378679362, −9.705693592679979661925070429855, −9.277967185900813285411063317474, −8.164895386292406078619635083769, −7.72407337362098831928992378282, −6.78031007921562599657583235141, −4.88311035495610326508509429800, −3.49389185098859385609006115477, −2.53173862327978090215295081586, −1.35664056388397445209219995392, 1.35664056388397445209219995392, 2.53173862327978090215295081586, 3.49389185098859385609006115477, 4.88311035495610326508509429800, 6.78031007921562599657583235141, 7.72407337362098831928992378282, 8.164895386292406078619635083769, 9.277967185900813285411063317474, 9.705693592679979661925070429855, 10.67578773904240973083378679362

Graph of the ZZ-function along the critical line