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Properties

Label 2-1911-1.1-c3-0-242
Degree 22
Conductor 19111911
Sign −1-1
Analytic cond. 112.752112.752
Root an. cond. 10.618510.6185
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.74·2-s + 3·3-s + 14.4·4-s − 4.51·5-s + 14.2·6-s + 30.7·8-s + 9·9-s − 21.4·10-s − 66.8·11-s + 43.4·12-s + 13·13-s − 13.5·15-s + 29.8·16-s − 96.9·17-s + 42.6·18-s − 31.4·19-s − 65.4·20-s − 317.·22-s + 183.·23-s + 92.2·24-s − 104.·25-s + 61.6·26-s + 27·27-s + 112.·29-s − 64.2·30-s + 77.2·31-s − 104.·32-s + ⋯
L(s)  = 1  + 1.67·2-s + 0.577·3-s + 1.81·4-s − 0.403·5-s + 0.967·6-s + 1.35·8-s + 0.333·9-s − 0.677·10-s − 1.83·11-s + 1.04·12-s + 0.277·13-s − 0.233·15-s + 0.467·16-s − 1.38·17-s + 0.558·18-s − 0.380·19-s − 0.731·20-s − 3.07·22-s + 1.66·23-s + 0.784·24-s − 0.836·25-s + 0.464·26-s + 0.192·27-s + 0.718·29-s − 0.391·30-s + 0.447·31-s − 0.575·32-s + ⋯

Functional equation

Λ(s)=(1911s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1911s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19111911    =    3⋅72⋅133 \cdot 7^{2} \cdot 13
Sign: −1-1
Analytic conductor: 112.752112.752
Root analytic conductor: 10.618510.6185
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1911, ( :3/2), −1)(2,\ 1911,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1−3T 1 - 3T
7 1 1
13 1−13T 1 - 13T
good2 1−4.74T+8T2 1 - 4.74T + 8T^{2}
5 1+4.51T+125T2 1 + 4.51T + 125T^{2}
11 1+66.8T+1.33e3T2 1 + 66.8T + 1.33e3T^{2}
17 1+96.9T+4.91e3T2 1 + 96.9T + 4.91e3T^{2}
19 1+31.4T+6.85e3T2 1 + 31.4T + 6.85e3T^{2}
23 1−183.T+1.21e4T2 1 - 183.T + 1.21e4T^{2}
29 1−112.T+2.43e4T2 1 - 112.T + 2.43e4T^{2}
31 1−77.2T+2.97e4T2 1 - 77.2T + 2.97e4T^{2}
37 1−54.7T+5.06e4T2 1 - 54.7T + 5.06e4T^{2}
41 1+451.T+6.89e4T2 1 + 451.T + 6.89e4T^{2}
43 1+113.T+7.95e4T2 1 + 113.T + 7.95e4T^{2}
47 1−42.2T+1.03e5T2 1 - 42.2T + 1.03e5T^{2}
53 1+530.T+1.48e5T2 1 + 530.T + 1.48e5T^{2}
59 1+219.T+2.05e5T2 1 + 219.T + 2.05e5T^{2}
61 1+822.T+2.26e5T2 1 + 822.T + 2.26e5T^{2}
67 1+872.T+3.00e5T2 1 + 872.T + 3.00e5T^{2}
71 1+100.T+3.57e5T2 1 + 100.T + 3.57e5T^{2}
73 1−165.T+3.89e5T2 1 - 165.T + 3.89e5T^{2}
79 1+545.T+4.93e5T2 1 + 545.T + 4.93e5T^{2}
83 1−454.T+5.71e5T2 1 - 454.T + 5.71e5T^{2}
89 1−230.T+7.04e5T2 1 - 230.T + 7.04e5T^{2}
97 1−1.08e3T+9.12e5T2 1 - 1.08e3T + 9.12e5T^{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

−8.288347183102860276836602310675, −7.49523734192482130163724293757, −6.72643995566860476874215053249, −5.91051238360503669601829896524, −4.77227033238490162556810899645, −4.62588227147322334931894378046, −3.32870206181933918744786977180, −2.83422529011940116004151051014, −1.90348968547955548894483240166, 0, 1.90348968547955548894483240166, 2.83422529011940116004151051014, 3.32870206181933918744786977180, 4.62588227147322334931894378046, 4.77227033238490162556810899645, 5.91051238360503669601829896524, 6.72643995566860476874215053249, 7.49523734192482130163724293757, 8.288347183102860276836602310675

Graph of the ZZ-function along the critical line