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Properties

Label 2-350-1.1-c1-0-9
Degree 22
Conductor 350350
Sign −1-1
Analytic cond. 2.794762.79476
Root an. cond. 1.671751.67175
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 3·6-s − 7-s + 8-s + 6·9-s − 5·11-s − 3·12-s − 6·13-s − 14-s + 16-s − 17-s + 6·18-s − 3·19-s + 3·21-s − 5·22-s − 3·24-s − 6·26-s − 9·27-s − 28-s − 6·29-s − 4·31-s + 32-s + 15·33-s − 34-s + 6·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.50·11-s − 0.866·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.41·18-s − 0.688·19-s + 0.654·21-s − 1.06·22-s − 0.612·24-s − 1.17·26-s − 1.73·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 2.61·33-s − 0.171·34-s + 36-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 2.794762.79476
Root analytic conductor: 1.671751.67175
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 350, ( :1/2), −1)(2,\ 350,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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−T 1 - T
5 1 1
7 1+T 1 + T
good3 1+pT+pT2 1 + p T + p T^{2}
11 1+5T+pT2 1 + 5 T + p T^{2}
13 1+6T+pT2 1 + 6 T + p T^{2}
17 1+T+pT2 1 + T + p T^{2}
19 1+3T+pT2 1 + 3 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−8T+pT2 1 - 8 T + p T^{2}
41 1−11T+pT2 1 - 11 T + p T^{2}
43 1+8T+pT2 1 + 8 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−9T+pT2 1 - 9 T + p T^{2}
71 1+10T+pT2 1 + 10 T + p T^{2}
73 1+7T+pT2 1 + 7 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−11T+pT2 1 - 11 T + p T^{2}
89 1+11T+pT2 1 + 11 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p 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

−11.11858258197447904087693203763, −10.45494407305386764464659959764, −9.627920882288908715420031969238, −7.70054391482804408966188304160, −6.95538626651508714624591948210, −5.84573096199937208498827543184, −5.20618442524944639616146822933, −4.32124854619714589271574708665, −2.44905781159727834481891285350, 0, 2.44905781159727834481891285350, 4.32124854619714589271574708665, 5.20618442524944639616146822933, 5.84573096199937208498827543184, 6.95538626651508714624591948210, 7.70054391482804408966188304160, 9.627920882288908715420031969238, 10.45494407305386764464659959764, 11.11858258197447904087693203763

Graph of the ZZ-function along the critical line