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Properties

Label 2-3131-3131.1115-c0-0-2
Degree $2$
Conductor $3131$
Sign $0.522 + 0.852i$
Analytic cond. $1.56257$
Root an. cond. $1.25002$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 0.123i)2-s + (2.83 + 0.358i)4-s + (−0.110 − 1.74i)5-s + (−1.65 − 1.05i)7-s + (3.58 + 0.684i)8-s + (−0.728 + 0.684i)9-s − 3.44i·10-s + (−3.12 − 2.26i)14-s + (4.18 + 1.07i)16-s + (−1.51 + 1.25i)18-s + (1.03 − 0.266i)19-s + (0.314 − 5.00i)20-s + (−2.05 + 0.259i)25-s + (−4.32 − 3.58i)28-s + (−0.535 + 0.844i)31-s + (4.59 + 1.49i)32-s + ⋯
L(s)  = 1  + (1.96 + 0.123i)2-s + (2.83 + 0.358i)4-s + (−0.110 − 1.74i)5-s + (−1.65 − 1.05i)7-s + (3.58 + 0.684i)8-s + (−0.728 + 0.684i)9-s − 3.44i·10-s + (−3.12 − 2.26i)14-s + (4.18 + 1.07i)16-s + (−1.51 + 1.25i)18-s + (1.03 − 0.266i)19-s + (0.314 − 5.00i)20-s + (−2.05 + 0.259i)25-s + (−4.32 − 3.58i)28-s + (−0.535 + 0.844i)31-s + (4.59 + 1.49i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3131 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3131\)    =    \(31 \cdot 101\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(1.56257\)
Root analytic conductor: \(1.25002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3131} (1115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3131,\ (\ :0),\ 0.522 + 0.852i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.542456523\)
\(L(\frac12)\) \(\approx\) \(3.542456523\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad31 \( 1 + (0.535 - 0.844i)T \)
101 \( 1 + (0.0627 + 0.998i)T \)
good2 \( 1 + (-1.96 - 0.123i)T + (0.992 + 0.125i)T^{2} \)
3 \( 1 + (0.728 - 0.684i)T^{2} \)
5 \( 1 + (0.110 + 1.74i)T + (-0.992 + 0.125i)T^{2} \)
7 \( 1 + (1.65 + 1.05i)T + (0.425 + 0.904i)T^{2} \)
11 \( 1 + (0.0627 - 0.998i)T^{2} \)
13 \( 1 + (0.425 - 0.904i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-1.03 + 0.266i)T + (0.876 - 0.481i)T^{2} \)
23 \( 1 + (0.187 + 0.982i)T^{2} \)
29 \( 1 + (-0.425 + 0.904i)T^{2} \)
37 \( 1 + (-0.728 - 0.684i)T^{2} \)
41 \( 1 + (-1.72 + 0.559i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 + (-0.535 + 0.844i)T^{2} \)
47 \( 1 + (-0.541 - 0.297i)T + (0.535 + 0.844i)T^{2} \)
53 \( 1 + (0.968 - 0.248i)T^{2} \)
59 \( 1 + (0.183 - 0.713i)T + (-0.876 - 0.481i)T^{2} \)
61 \( 1 + (0.968 + 0.248i)T^{2} \)
67 \( 1 + (0.183 - 0.462i)T + (-0.728 - 0.684i)T^{2} \)
71 \( 1 + (1.80 - 0.713i)T + (0.728 - 0.684i)T^{2} \)
73 \( 1 + (-0.187 - 0.982i)T^{2} \)
79 \( 1 + (0.187 - 0.982i)T^{2} \)
83 \( 1 + (-0.187 + 0.982i)T^{2} \)
89 \( 1 + (0.876 - 0.481i)T^{2} \)
97 \( 1 + (-0.613 - 0.0774i)T + (0.968 + 0.248i)T^{2} \)
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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

−8.639455311285044125174801686678, −7.51840157167794861341620279950, −7.21406522430274627510434919081, −6.02462560725733993193124876313, −5.64534606175699633097701670075, −4.80652461409390225319998161197, −4.18532512970252498462263951829, −3.43757346184757760183906856641, −2.61848103340843320131994800629, −1.17424969983168000402341846893, 2.31115366993456832335702376399, 2.90787430952779997385723988410, 3.31921722328500020127455278786, 3.96261544632995524756727674154, 5.44033574373427428027932638249, 6.06714766905854173526540675501, 6.28996788979529761238270367620, 7.05455825604168105629390158829, 7.71016925346483048702822876242, 9.266294678984281133432090574185

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