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Properties

Label 2-7800-1.1-c1-0-34
Degree 22
Conductor 78007800
Sign 11
Analytic cond. 62.283362.2833
Root an. cond. 7.891977.89197
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2·7-s + 9-s − 13-s + 4·17-s + 6·19-s − 2·21-s − 6·23-s − 27-s + 4·29-s + 8·31-s + 6·37-s + 39-s + 6·41-s − 4·43-s − 8·47-s − 3·49-s − 4·51-s − 2·53-s − 6·57-s − 2·61-s + 2·63-s + 4·67-s + 6·69-s − 8·71-s + 16·79-s + 81-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.755·7-s + 1/3·9-s − 0.277·13-s + 0.970·17-s + 1.37·19-s − 0.436·21-s − 1.25·23-s − 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.986·37-s + 0.160·39-s + 0.937·41-s − 0.609·43-s − 1.16·47-s − 3/7·49-s − 0.560·51-s − 0.274·53-s − 0.794·57-s − 0.256·61-s + 0.251·63-s + 0.488·67-s + 0.722·69-s − 0.949·71-s + 1.80·79-s + 1/9·81-s + ⋯

Functional equation

Λ(s)=(7800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 78007800    =    23⋅3⋅52⋅132^{3} \cdot 3 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 62.283362.2833
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7800, ( :1/2), 1)(2,\ 7800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.9782134711.978213471
L(12)L(\frac12) ≈\approx 1.9782134711.978213471
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 1
3 1+T 1 + T
5 1 1
13 1+T 1 + T
good7 1−2T+pT2 1 - 2 T + p T^{2}
11 1+pT2 1 + p T^{2}
17 1−4T+pT2 1 - 4 T + p T^{2}
19 1−6T+pT2 1 - 6 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−6T+pT2 1 - 6 T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1+8T+pT2 1 + 8 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1+pT2 1 + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−4T+pT2 1 - 4 T + p T^{2}
71 1+8T+pT2 1 + 8 T + p T^{2}
73 1+pT2 1 + p T^{2}
79 1−16T+pT2 1 - 16 T + p T^{2}
83 1−4T+pT2 1 - 4 T + p T^{2}
89 1+6T+pT2 1 + 6 T + p T^{2}
97 1+16T+pT2 1 + 16 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

−7.986622336300113308442054608512, −7.22795569086376819706411624696, −6.33949481008680480500487088605, −5.79600071764111841459942651372, −4.97946564076675613100018722733, −4.55018293538489446400556372225, −3.54791455598267040520875396042, −2.68727288426056290994599474575, −1.58864990401304168322356395338, −0.77106860614917570896246748945, 0.77106860614917570896246748945, 1.58864990401304168322356395338, 2.68727288426056290994599474575, 3.54791455598267040520875396042, 4.55018293538489446400556372225, 4.97946564076675613100018722733, 5.79600071764111841459942651372, 6.33949481008680480500487088605, 7.22795569086376819706411624696, 7.986622336300113308442054608512

Graph of the ZZ-function along the critical line