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Properties

Label 2-3822-1.1-c1-0-13
Degree 22
Conductor 38223822
Sign 11
Analytic cond. 30.518830.5188
Root an. cond. 5.524385.52438
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  − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s − 3·11-s + 12-s + 13-s − 15-s + 16-s + 2·17-s − 18-s − 20-s + 3·22-s + 6·23-s − 24-s − 4·25-s − 26-s + 27-s + 9·29-s + 30-s − 5·31-s − 32-s − 3·33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.639·22-s + 1.25·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 1.67·29-s + 0.182·30-s − 0.898·31-s − 0.176·32-s − 0.522·33-s + ⋯

Functional equation

Λ(s)=(3822s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(3822s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38223822    =    2⋅3⋅72⋅132 \cdot 3 \cdot 7^{2} \cdot 13
Sign: 11
Analytic conductor: 30.518830.5188
Root analytic conductor: 5.524385.52438
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 3822, ( :1/2), 1)(2,\ 3822,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.4125871261.412587126
L(12)L(\frac12) ≈\approx 1.4125871261.412587126
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
3 1−T 1 - T
7 1 1
13 1−T 1 - T
good5 1+T+pT2 1 + T + p T^{2}
11 1+3T+pT2 1 + 3 T + p T^{2}
17 1−2T+pT2 1 - 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1−9T+pT2 1 - 9 T + p T^{2}
31 1+5T+pT2 1 + 5 T + p T^{2}
37 1+8T+pT2 1 + 8 T + p T^{2}
41 1−4T+pT2 1 - 4 T + p T^{2}
43 1+pT2 1 + p T^{2}
47 1+pT2 1 + p T^{2}
53 1−T+pT2 1 - T + p T^{2}
59 1−7T+pT2 1 - 7 T + p T^{2}
61 1−4T+pT2 1 - 4 T + p T^{2}
67 1+4T+pT2 1 + 4 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+6T+pT2 1 + 6 T + p T^{2}
79 1+13T+pT2 1 + 13 T + p T^{2}
83 1−3T+pT2 1 - 3 T + p T^{2}
89 1−8T+pT2 1 - 8 T + p T^{2}
97 1−15T+pT2 1 - 15 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

−8.681317376442738162759058241445, −7.71697835609979599593253841498, −7.38708268557686291596612228621, −6.50197084254259805345325655638, −5.52888803600150368937602175554, −4.70472973283752392095637994703, −3.60037583418279129967512627544, −2.93242419346131617376141033463, −1.96369416177934919198686589063, −0.74015152998151316997665046224, 0.74015152998151316997665046224, 1.96369416177934919198686589063, 2.93242419346131617376141033463, 3.60037583418279129967512627544, 4.70472973283752392095637994703, 5.52888803600150368937602175554, 6.50197084254259805345325655638, 7.38708268557686291596612228621, 7.71697835609979599593253841498, 8.681317376442738162759058241445

Graph of the ZZ-function along the critical line