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Properties

Label 2-1881-1.1-c1-0-39
Degree 22
Conductor 18811881
Sign −1-1
Analytic cond. 15.019815.0198
Root an. cond. 3.875543.87554
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + 5-s − 3.41·7-s + 2.82·8-s − 1.41·10-s + 11-s + 2.24·13-s + 4.82·14-s − 4.00·16-s − 3.41·17-s − 19-s − 1.41·22-s + 3·23-s − 4·25-s − 3.17·26-s + 6.24·29-s − 6.41·31-s + 4.82·34-s − 3.41·35-s + 10.0·37-s + 1.41·38-s + 2.82·40-s + 1.65·41-s + 0.343·43-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.447·5-s − 1.29·7-s + 0.999·8-s − 0.447·10-s + 0.301·11-s + 0.621·13-s + 1.29·14-s − 1.00·16-s − 0.828·17-s − 0.229·19-s − 0.301·22-s + 0.625·23-s − 0.800·25-s − 0.621·26-s + 1.15·29-s − 1.15·31-s + 0.828·34-s − 0.577·35-s + 1.65·37-s + 0.229·38-s + 0.447·40-s + 0.258·41-s + 0.0523·43-s + ⋯

Functional equation

Λ(s)=(1881s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1881s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18811881    =    32⋅11⋅193^{2} \cdot 11 \cdot 19
Sign: −1-1
Analytic conductor: 15.019815.0198
Root analytic conductor: 3.875543.87554
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1881, ( :1/2), −1)(2,\ 1881,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
11 1−T 1 - T
19 1+T 1 + T
good2 1+1.41T+2T2 1 + 1.41T + 2T^{2}
5 1−T+5T2 1 - T + 5T^{2}
7 1+3.41T+7T2 1 + 3.41T + 7T^{2}
13 1−2.24T+13T2 1 - 2.24T + 13T^{2}
17 1+3.41T+17T2 1 + 3.41T + 17T^{2}
23 1−3T+23T2 1 - 3T + 23T^{2}
29 1−6.24T+29T2 1 - 6.24T + 29T^{2}
31 1+6.41T+31T2 1 + 6.41T + 31T^{2}
37 1−10.0T+37T2 1 - 10.0T + 37T^{2}
41 1−1.65T+41T2 1 - 1.65T + 41T^{2}
43 1−0.343T+43T2 1 - 0.343T + 43T^{2}
47 1+8.82T+47T2 1 + 8.82T + 47T^{2}
53 1−4.48T+53T2 1 - 4.48T + 53T^{2}
59 1−1.58T+59T2 1 - 1.58T + 59T^{2}
61 1+11.0T+61T2 1 + 11.0T + 61T^{2}
67 1+10.4T+67T2 1 + 10.4T + 67T^{2}
71 1−12.4T+71T2 1 - 12.4T + 71T^{2}
73 1+4.48T+73T2 1 + 4.48T + 73T^{2}
79 1+14.5T+79T2 1 + 14.5T + 79T^{2}
83 1+3.41T+83T2 1 + 3.41T + 83T^{2}
89 1+4.89T+89T2 1 + 4.89T + 89T^{2}
97 1−2.41T+97T2 1 - 2.41T + 97T^{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

−9.029616056575936623640895012181, −8.288545594411779593161493154789, −7.32557306461599310872818866419, −6.53131322980631352966087170390, −5.91388422041459863675062062540, −4.65982749344289371084893169663, −3.77064372604746606096012529394, −2.61645048657172399571406500208, −1.34977452635185729534176938195, 0, 1.34977452635185729534176938195, 2.61645048657172399571406500208, 3.77064372604746606096012529394, 4.65982749344289371084893169663, 5.91388422041459863675062062540, 6.53131322980631352966087170390, 7.32557306461599310872818866419, 8.288545594411779593161493154789, 9.029616056575936623640895012181

Graph of the ZZ-function along the critical line