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Properties

Label 2-1057-1057.195-c0-0-0
Degree $2$
Conductor $1057$
Sign $-0.752 + 0.658i$
Analytic cond. $0.527511$
Root an. cond. $0.726300$
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  + (−0.866 − 0.629i)2-s + (0.0458 + 0.141i)4-s + (0.0627 − 0.998i)7-s + (−0.282 + 0.867i)8-s + (−0.637 + 0.770i)9-s + (0.688 − 1.46i)11-s + (−0.683 + 0.825i)14-s + (0.911 − 0.662i)16-s + (1.03 − 0.266i)18-s + (−1.51 + 0.835i)22-s + (−0.5 − 1.53i)23-s + (−0.187 − 0.982i)25-s + (0.143 − 0.0369i)28-s + (−1.73 + 0.955i)29-s − 0.294·32-s + ⋯
L(s)  = 1  + (−0.866 − 0.629i)2-s + (0.0458 + 0.141i)4-s + (0.0627 − 0.998i)7-s + (−0.282 + 0.867i)8-s + (−0.637 + 0.770i)9-s + (0.688 − 1.46i)11-s + (−0.683 + 0.825i)14-s + (0.911 − 0.662i)16-s + (1.03 − 0.266i)18-s + (−1.51 + 0.835i)22-s + (−0.5 − 1.53i)23-s + (−0.187 − 0.982i)25-s + (0.143 − 0.0369i)28-s + (−1.73 + 0.955i)29-s − 0.294·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.752 + 0.658i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ -0.752 + 0.658i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5215234975\)
\(L(\frac12)\) \(\approx\) \(0.5215234975\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.0627 + 0.998i)T \)
151 \( 1 + (-0.968 + 0.248i)T \)
good2 \( 1 + (0.866 + 0.629i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (0.637 - 0.770i)T^{2} \)
5 \( 1 + (0.187 + 0.982i)T^{2} \)
11 \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \)
13 \( 1 + (-0.968 + 0.248i)T^{2} \)
17 \( 1 + (0.992 - 0.125i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.73 - 0.955i)T + (0.535 - 0.844i)T^{2} \)
31 \( 1 + (0.425 + 0.904i)T^{2} \)
37 \( 1 + (-1.60 + 0.202i)T + (0.968 - 0.248i)T^{2} \)
41 \( 1 + (0.929 + 0.368i)T^{2} \)
43 \( 1 + (0.0534 + 0.849i)T + (-0.992 + 0.125i)T^{2} \)
47 \( 1 + (0.929 + 0.368i)T^{2} \)
53 \( 1 + (0.273 + 0.256i)T + (0.0627 + 0.998i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.637 + 0.770i)T^{2} \)
67 \( 1 + (1.23 + 1.49i)T + (-0.187 + 0.982i)T^{2} \)
71 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
73 \( 1 + (0.992 + 0.125i)T^{2} \)
79 \( 1 + (-0.574 - 0.904i)T + (-0.425 + 0.904i)T^{2} \)
83 \( 1 + (-0.728 + 0.684i)T^{2} \)
89 \( 1 + (-0.728 + 0.684i)T^{2} \)
97 \( 1 + (0.187 - 0.982i)T^{2} \)
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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

−9.897362284487793039979081958032, −8.934063646673173698705299038920, −8.369546588204757858532783245644, −7.64410411668395796450403010486, −6.36671868048335425426775412090, −5.59171726614522922618489779165, −4.40610243029548672061752703008, −3.25083726726205392278410362219, −2.07116501234742544196504656380, −0.64950637308375359569075091734, 1.75646113122650439240349456618, 3.24414983252721661873686749298, 4.26192710140281726064580612643, 5.67927764765735805343302496397, 6.26716798116560498846319726366, 7.34878299970040396679776128868, 7.84848954041839285824555054776, 8.969832753285570421431529047080, 9.444159227189174882168388417406, 9.789856957482557015753928795886

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