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Properties

Label 2-7800-1.1-c1-0-18
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 + 9-s − 4·11-s − 13-s − 2·17-s − 4·19-s − 8·23-s + 27-s + 6·29-s + 8·31-s − 4·33-s − 6·37-s − 39-s + 2·41-s + 12·43-s + 8·47-s − 7·49-s − 2·51-s + 2·53-s − 4·57-s + 12·59-s − 2·61-s − 12·67-s − 8·69-s + 16·71-s − 10·73-s + 8·79-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s − 0.160·39-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 0.280·51-s + 0.274·53-s − 0.529·57-s + 1.56·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + 1.89·71-s − 1.17·73-s + 0.900·79-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 1.9178196701.917819670
L(12)L(\frac12) ≈\approx 1.9178196701.917819670
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+pT2 1 + p T^{2}
11 1+4T+pT2 1 + 4 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+8T+pT2 1 + 8 T + p T^{2}
29 1−6T+pT2 1 - 6 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 1−2T+pT2 1 - 2 T + p T^{2}
43 1−12T+pT2 1 - 12 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 1−12T+pT2 1 - 12 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1+12T+pT2 1 + 12 T + p T^{2}
71 1−16T+pT2 1 - 16 T + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−8T+pT2 1 - 8 T + p T^{2}
83 1−12T+pT2 1 - 12 T + p T^{2}
89 1−18T+pT2 1 - 18 T + p T^{2}
97 1+2T+pT2 1 + 2 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

−7.939428236970305679916521566968, −7.33133212186291559292656053100, −6.43524788089026139039377475493, −5.88592241444235658101323561619, −4.86565535063385090482663143888, −4.36054460607134966378609243541, −3.48486855853978324398418219304, −2.46066849465400339563544193095, −2.15821041599584095280752391820, −0.63876466895030982743652561062, 0.63876466895030982743652561062, 2.15821041599584095280752391820, 2.46066849465400339563544193095, 3.48486855853978324398418219304, 4.36054460607134966378609243541, 4.86565535063385090482663143888, 5.88592241444235658101323561619, 6.43524788089026139039377475493, 7.33133212186291559292656053100, 7.939428236970305679916521566968

Graph of the ZZ-function along the critical line