Properties

Label 2-136710-1.1-c1-0-117
Degree $2$
Conductor $136710$
Sign $-1$
Analytic cond. $1091.63$
Root an. cond. $33.0398$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s − 6·13-s + 16-s + 2·17-s − 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s + 6·26-s − 6·29-s + 31-s − 32-s − 2·34-s − 2·37-s + 4·38-s − 40-s + 10·41-s − 4·43-s + 4·44-s − 8·46-s − 50-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s − 0.141·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136710\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 31\)
Sign: $-1$
Analytic conductor: \(1091.63\)
Root analytic conductor: \(33.0398\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 136710,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
31 \( 1 - T \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.68857106868605, −13.10818724587253, −12.58096893133845, −12.22598143890069, −11.78756756616428, −11.09163315739443, −10.84267768104473, −10.20124840078902, −9.672072172963575, −9.343927485061639, −8.977980584854504, −8.441195284723397, −7.711863831874515, −7.321503742648290, −6.780151943441869, −6.510277121306070, −5.575022248840204, −5.428247166800758, −4.485928433887297, −4.180397865538745, −3.239239904931330, −2.768692476539447, −2.115377471751269, −1.530186622159863, −0.8564640589477164, 0, 0.8564640589477164, 1.530186622159863, 2.115377471751269, 2.768692476539447, 3.239239904931330, 4.180397865538745, 4.485928433887297, 5.428247166800758, 5.575022248840204, 6.510277121306070, 6.780151943441869, 7.321503742648290, 7.711863831874515, 8.441195284723397, 8.977980584854504, 9.343927485061639, 9.672072172963575, 10.20124840078902, 10.84267768104473, 11.09163315739443, 11.78756756616428, 12.22598143890069, 12.58096893133845, 13.10818724587253, 13.68857106868605

Graph of the $Z$-function along the critical line