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Properties

Label 2-350-1.1-c3-0-22
Degree 22
Conductor 350350
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 4·3-s + 4·4-s − 8·6-s − 7·7-s + 8·8-s − 11·9-s + 60·11-s − 16·12-s − 38·13-s − 14·14-s + 16·16-s − 42·17-s − 22·18-s − 52·19-s + 28·21-s + 120·22-s − 120·23-s − 32·24-s − 76·26-s + 152·27-s − 28·28-s − 234·29-s − 304·31-s + 32·32-s − 240·33-s − 84·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s − 0.377·7-s + 0.353·8-s − 0.407·9-s + 1.64·11-s − 0.384·12-s − 0.810·13-s − 0.267·14-s + 1/4·16-s − 0.599·17-s − 0.288·18-s − 0.627·19-s + 0.290·21-s + 1.16·22-s − 1.08·23-s − 0.272·24-s − 0.573·26-s + 1.08·27-s − 0.188·28-s − 1.49·29-s − 1.76·31-s + 0.176·32-s − 1.26·33-s − 0.423·34-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: −1-1
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: 11
Selberg data: (2, 350, ( :3/2), −1)(2,\ 350,\ (\ :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}
ppFp(T)F_p(T)
bad2 1−pT 1 - p T
5 1 1
7 1+pT 1 + p T
good3 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1−60T+p3T2 1 - 60 T + p^{3} T^{2}
13 1+38T+p3T2 1 + 38 T + p^{3} T^{2}
17 1+42T+p3T2 1 + 42 T + p^{3} T^{2}
19 1+52T+p3T2 1 + 52 T + p^{3} T^{2}
23 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
29 1+234T+p3T2 1 + 234 T + p^{3} T^{2}
31 1+304T+p3T2 1 + 304 T + p^{3} T^{2}
37 1−106T+p3T2 1 - 106 T + p^{3} T^{2}
41 1+54T+p3T2 1 + 54 T + p^{3} T^{2}
43 1−196T+p3T2 1 - 196 T + p^{3} T^{2}
47 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
53 1+438T+p3T2 1 + 438 T + p^{3} T^{2}
59 1+444T+p3T2 1 + 444 T + p^{3} T^{2}
61 1−38T+p3T2 1 - 38 T + p^{3} T^{2}
67 1−988T+p3T2 1 - 988 T + p^{3} T^{2}
71 1+720T+p3T2 1 + 720 T + p^{3} T^{2}
73 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
79 1+808T+p3T2 1 + 808 T + p^{3} T^{2}
83 1+612T+p3T2 1 + 612 T + p^{3} T^{2}
89 1−1146T+p3T2 1 - 1146 T + p^{3} T^{2}
97 1−70T+p3T2 1 - 70 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

−11.01134479195513610671450819676, −9.750552157819167196193556983835, −8.859944742364187782911867263387, −7.39210064155521959827008538110, −6.42049730348402531031518873779, −5.79182796448254136711944431958, −4.56583699129886552541421384600, −3.55767192260810315828251616378, −1.93982667360229045690498247808, 0, 1.93982667360229045690498247808, 3.55767192260810315828251616378, 4.56583699129886552541421384600, 5.79182796448254136711944431958, 6.42049730348402531031518873779, 7.39210064155521959827008538110, 8.859944742364187782911867263387, 9.750552157819167196193556983835, 11.01134479195513610671450819676

Graph of the ZZ-function along the critical line