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Properties

Label 2-6336-1.1-c1-0-23
Degree 22
Conductor 63366336
Sign 11
Analytic cond. 50.593250.5932
Root an. cond. 7.112897.11289
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  − 3·5-s + 2·7-s + 11-s + 6·17-s + 4·19-s + 23-s + 4·25-s − 8·29-s + 7·31-s − 6·35-s + 37-s − 4·41-s + 6·43-s − 8·47-s − 3·49-s + 2·53-s − 3·55-s + 59-s − 4·61-s − 5·67-s + 3·71-s + 16·73-s + 2·77-s − 2·79-s + 2·83-s − 18·85-s − 15·89-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.755·7-s + 0.301·11-s + 1.45·17-s + 0.917·19-s + 0.208·23-s + 4/5·25-s − 1.48·29-s + 1.25·31-s − 1.01·35-s + 0.164·37-s − 0.624·41-s + 0.914·43-s − 1.16·47-s − 3/7·49-s + 0.274·53-s − 0.404·55-s + 0.130·59-s − 0.512·61-s − 0.610·67-s + 0.356·71-s + 1.87·73-s + 0.227·77-s − 0.225·79-s + 0.219·83-s − 1.95·85-s − 1.58·89-s + ⋯

Functional equation

Λ(s)=(6336s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6336s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63366336    =    26⋅32⋅112^{6} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 50.593250.5932
Root analytic conductor: 7.112897.11289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6336, ( :1/2), 1)(2,\ 6336,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.7360879121.736087912
L(12)L(\frac12) ≈\approx 1.7360879121.736087912
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 1
11 1−T 1 - T
good5 1+3T+pT2 1 + 3 T + p T^{2}
7 1−2T+pT2 1 - 2 T + p T^{2}
13 1+pT2 1 + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−4T+pT2 1 - 4 T + p T^{2}
23 1−T+pT2 1 - T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1−7T+pT2 1 - 7 T + p T^{2}
37 1−T+pT2 1 - T + p T^{2}
41 1+4T+pT2 1 + 4 T + p T^{2}
43 1−6T+pT2 1 - 6 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−T+pT2 1 - T + p T^{2}
61 1+4T+pT2 1 + 4 T + p T^{2}
67 1+5T+pT2 1 + 5 T + p T^{2}
71 1−3T+pT2 1 - 3 T + p T^{2}
73 1−16T+pT2 1 - 16 T + p T^{2}
79 1+2T+pT2 1 + 2 T + p T^{2}
83 1−2T+pT2 1 - 2 T + p T^{2}
89 1+15T+pT2 1 + 15 T + p T^{2}
97 1+7T+pT2 1 + 7 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

−7.927629123853046774697751456251, −7.55252528081176974112132450170, −6.82671507782272554192070208796, −5.79242088578546656641015440150, −5.10583765748157113159455391169, −4.37221035310191883411362450637, −3.60822760858280223727828451956, −3.02374111947685053319903637317, −1.66294893612641948046386286042, −0.72016671080523259380610994522, 0.72016671080523259380610994522, 1.66294893612641948046386286042, 3.02374111947685053319903637317, 3.60822760858280223727828451956, 4.37221035310191883411362450637, 5.10583765748157113159455391169, 5.79242088578546656641015440150, 6.82671507782272554192070208796, 7.55252528081176974112132450170, 7.927629123853046774697751456251

Graph of the ZZ-function along the critical line