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Properties

Label 2-12e3-1.1-c3-0-19
Degree 22
Conductor 17281728
Sign 11
Analytic cond. 101.955101.955
Root an. cond. 10.097210.0972
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15·5-s − 25·7-s − 15·11-s − 20·13-s − 72·17-s − 2·19-s − 114·23-s + 100·25-s + 30·29-s + 101·31-s − 375·35-s + 430·37-s + 30·41-s − 110·43-s + 330·47-s + 282·49-s + 621·53-s − 225·55-s − 660·59-s + 376·61-s − 300·65-s + 250·67-s + 360·71-s + 785·73-s + 375·77-s + 488·79-s + 489·83-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.34·7-s − 0.411·11-s − 0.426·13-s − 1.02·17-s − 0.0241·19-s − 1.03·23-s + 4/5·25-s + 0.192·29-s + 0.585·31-s − 1.81·35-s + 1.91·37-s + 0.114·41-s − 0.390·43-s + 1.02·47-s + 0.822·49-s + 1.60·53-s − 0.551·55-s − 1.45·59-s + 0.789·61-s − 0.572·65-s + 0.455·67-s + 0.601·71-s + 1.25·73-s + 0.555·77-s + 0.694·79-s + 0.646·83-s + ⋯

Functional equation

Λ(s)=(1728s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1728s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17281728    =    26⋅332^{6} \cdot 3^{3}
Sign: 11
Analytic conductor: 101.955101.955
Root analytic conductor: 10.097210.0972
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1728, ( :3/2), 1)(2,\ 1728,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.8679638281.867963828
L(12)L(\frac12) ≈\approx 1.8679638281.867963828
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
good5 1−3pT+p3T2 1 - 3 p T + p^{3} T^{2}
7 1+25T+p3T2 1 + 25 T + p^{3} T^{2}
11 1+15T+p3T2 1 + 15 T + p^{3} T^{2}
13 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
17 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
19 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
23 1+114T+p3T2 1 + 114 T + p^{3} T^{2}
29 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
31 1−101T+p3T2 1 - 101 T + p^{3} T^{2}
37 1−430T+p3T2 1 - 430 T + p^{3} T^{2}
41 1−30T+p3T2 1 - 30 T + p^{3} T^{2}
43 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
47 1−330T+p3T2 1 - 330 T + p^{3} T^{2}
53 1−621T+p3T2 1 - 621 T + p^{3} T^{2}
59 1+660T+p3T2 1 + 660 T + p^{3} T^{2}
61 1−376T+p3T2 1 - 376 T + p^{3} T^{2}
67 1−250T+p3T2 1 - 250 T + p^{3} T^{2}
71 1−360T+p3T2 1 - 360 T + p^{3} T^{2}
73 1−785T+p3T2 1 - 785 T + p^{3} T^{2}
79 1−488T+p3T2 1 - 488 T + p^{3} T^{2}
83 1−489T+p3T2 1 - 489 T + p^{3} T^{2}
89 1−450T+p3T2 1 - 450 T + p^{3} T^{2}
97 1+1105T+p3T2 1 + 1105 T + p^{3} 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

−9.247204458725974793416278544461, −8.266769664436970503407941807189, −7.20486625028901444291783322961, −6.30795345537184846933462995507, −6.00843632866330852465018389541, −4.96563000226933331029469000817, −3.90561742116063638496522847829, −2.67881220184090618468696904049, −2.16069670954500208485082317390, −0.61855290661416154441323143999, 0.61855290661416154441323143999, 2.16069670954500208485082317390, 2.67881220184090618468696904049, 3.90561742116063638496522847829, 4.96563000226933331029469000817, 6.00843632866330852465018389541, 6.30795345537184846933462995507, 7.20486625028901444291783322961, 8.266769664436970503407941807189, 9.247204458725974793416278544461

Graph of the ZZ-function along the critical line