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Properties

Label 2-1872-1.1-c3-0-37
Degree 22
Conductor 18721872
Sign −1-1
Analytic cond. 110.451110.451
Root an. cond. 10.509510.5095
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  − 19.1·5-s − 35.1·7-s + 26·11-s − 13·13-s + 36.2·17-s + 95.5·19-s − 161.·23-s + 240.·25-s + 91.3·29-s + 266.·31-s + 671.·35-s − 149.·37-s + 77.8·41-s − 183.·43-s − 60.6·47-s + 890.·49-s − 281.·53-s − 496.·55-s − 542.·59-s + 65.0·61-s + 248.·65-s + 1.03e3·67-s + 1.04e3·71-s + 483.·73-s − 912.·77-s + 1.33e3·79-s + 812.·83-s + ⋯
L(s)  = 1  − 1.70·5-s − 1.89·7-s + 0.712·11-s − 0.277·13-s + 0.516·17-s + 1.15·19-s − 1.46·23-s + 1.92·25-s + 0.585·29-s + 1.54·31-s + 3.24·35-s − 0.664·37-s + 0.296·41-s − 0.651·43-s − 0.188·47-s + 2.59·49-s − 0.729·53-s − 1.21·55-s − 1.19·59-s + 0.136·61-s + 0.474·65-s + 1.88·67-s + 1.74·71-s + 0.774·73-s − 1.35·77-s + 1.90·79-s + 1.07·83-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 110.451110.451
Root analytic conductor: 10.509510.5095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1872, ( :3/2), −1)(2,\ 1872,\ (\ :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)
bad2 1 1
3 1 1
13 1+13T 1 + 13T
good5 1+19.1T+125T2 1 + 19.1T + 125T^{2}
7 1+35.1T+343T2 1 + 35.1T + 343T^{2}
11 1−26T+1.33e3T2 1 - 26T + 1.33e3T^{2}
17 1−36.2T+4.91e3T2 1 - 36.2T + 4.91e3T^{2}
19 1−95.5T+6.85e3T2 1 - 95.5T + 6.85e3T^{2}
23 1+161.T+1.21e4T2 1 + 161.T + 1.21e4T^{2}
29 1−91.3T+2.43e4T2 1 - 91.3T + 2.43e4T^{2}
31 1−266.T+2.97e4T2 1 - 266.T + 2.97e4T^{2}
37 1+149.T+5.06e4T2 1 + 149.T + 5.06e4T^{2}
41 1−77.8T+6.89e4T2 1 - 77.8T + 6.89e4T^{2}
43 1+183.T+7.95e4T2 1 + 183.T + 7.95e4T^{2}
47 1+60.6T+1.03e5T2 1 + 60.6T + 1.03e5T^{2}
53 1+281.T+1.48e5T2 1 + 281.T + 1.48e5T^{2}
59 1+542.T+2.05e5T2 1 + 542.T + 2.05e5T^{2}
61 1−65.0T+2.26e5T2 1 - 65.0T + 2.26e5T^{2}
67 1−1.03e3T+3.00e5T2 1 - 1.03e3T + 3.00e5T^{2}
71 1−1.04e3T+3.57e5T2 1 - 1.04e3T + 3.57e5T^{2}
73 1−483.T+3.89e5T2 1 - 483.T + 3.89e5T^{2}
79 1−1.33e3T+4.93e5T2 1 - 1.33e3T + 4.93e5T^{2}
83 1−812.T+5.71e5T2 1 - 812.T + 5.71e5T^{2}
89 1+936.T+7.04e5T2 1 + 936.T + 7.04e5T^{2}
97 1+954.T+9.12e5T2 1 + 954.T + 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

−8.281070193100072734471404005118, −7.78710342068889981386247026438, −6.77124169218390644567232598628, −6.42864839786140833994736941152, −5.16319157197850804973390641030, −3.99440462185048265865660132166, −3.55414415699425161581413133450, −2.78038519231245257871463135579, −0.884414296797356043782937990923, 0, 0.884414296797356043782937990923, 2.78038519231245257871463135579, 3.55414415699425161581413133450, 3.99440462185048265865660132166, 5.16319157197850804973390641030, 6.42864839786140833994736941152, 6.77124169218390644567232598628, 7.78710342068889981386247026438, 8.281070193100072734471404005118

Graph of the ZZ-function along the critical line