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Properties

Label 2-234-1.1-c5-0-23
Degree 22
Conductor 234234
Sign −1-1
Analytic cond. 37.529837.5298
Root an. cond. 6.126156.12615
Motivic weight 55
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 14·5-s − 170·7-s + 64·8-s + 56·10-s + 250·11-s − 169·13-s − 680·14-s + 256·16-s − 1.06e3·17-s − 78·19-s + 224·20-s + 1.00e3·22-s − 1.57e3·23-s − 2.92e3·25-s − 676·26-s − 2.72e3·28-s − 2.57e3·29-s − 8.65e3·31-s + 1.02e3·32-s − 4.24e3·34-s − 2.38e3·35-s + 1.09e4·37-s − 312·38-s + 896·40-s − 1.05e3·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.250·5-s − 1.31·7-s + 0.353·8-s + 0.177·10-s + 0.622·11-s − 0.277·13-s − 0.927·14-s + 1/4·16-s − 0.891·17-s − 0.0495·19-s + 0.125·20-s + 0.440·22-s − 0.621·23-s − 0.937·25-s − 0.196·26-s − 0.655·28-s − 0.569·29-s − 1.61·31-s + 0.176·32-s − 0.630·34-s − 0.328·35-s + 1.31·37-s − 0.0350·38-s + 0.0885·40-s − 0.0975·41-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=(− Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+5/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: −1-1
Analytic conductor: 37.529837.5298
Root analytic conductor: 6.126156.12615
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 234, ( :5/2), −1)(2,\ 234,\ (\ :5/2),\ -1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{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−p2T 1 - p^{2} T
3 1 1
13 1+p2T 1 + p^{2} T
good5 1−14T+p5T2 1 - 14 T + p^{5} T^{2}
7 1+170T+p5T2 1 + 170 T + p^{5} T^{2}
11 1−250T+p5T2 1 - 250 T + p^{5} T^{2}
17 1+1062T+p5T2 1 + 1062 T + p^{5} T^{2}
19 1+78T+p5T2 1 + 78 T + p^{5} T^{2}
23 1+1576T+p5T2 1 + 1576 T + p^{5} T^{2}
29 1+2578T+p5T2 1 + 2578 T + p^{5} T^{2}
31 1+8654T+p5T2 1 + 8654 T + p^{5} T^{2}
37 1−10986T+p5T2 1 - 10986 T + p^{5} T^{2}
41 1+1050T+p5T2 1 + 1050 T + p^{5} T^{2}
43 1+5900T+p5T2 1 + 5900 T + p^{5} T^{2}
47 1−5962T+p5T2 1 - 5962 T + p^{5} T^{2}
53 1+29046T+p5T2 1 + 29046 T + p^{5} T^{2}
59 1−13922T+p5T2 1 - 13922 T + p^{5} T^{2}
61 1+32882T+p5T2 1 + 32882 T + p^{5} T^{2}
67 1+69566T+p5T2 1 + 69566 T + p^{5} T^{2}
71 1−50542T+p5T2 1 - 50542 T + p^{5} T^{2}
73 1+46750T+p5T2 1 + 46750 T + p^{5} T^{2}
79 1+19348T+p5T2 1 + 19348 T + p^{5} T^{2}
83 1−87438T+p5T2 1 - 87438 T + p^{5} T^{2}
89 1+94170T+p5T2 1 + 94170 T + p^{5} T^{2}
97 1−182786T+p5T2 1 - 182786 T + p^{5} 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

−10.96286446143680722125998040716, −9.811940174037741267880492204866, −9.110867397245520150626416888061, −7.55006747926477459292837023913, −6.50064778054188545329943562289, −5.80456699072340915757753756135, −4.31479327546449218348073586296, −3.29815951405891859265258186166, −1.96651068802625974033326410568, 0, 1.96651068802625974033326410568, 3.29815951405891859265258186166, 4.31479327546449218348073586296, 5.80456699072340915757753756135, 6.50064778054188545329943562289, 7.55006747926477459292837023913, 9.110867397245520150626416888061, 9.811940174037741267880492204866, 10.96286446143680722125998040716

Graph of the ZZ-function along the critical line