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Properties

Label 2-1057-1057.20-c0-0-0
Degree $2$
Conductor $1057$
Sign $-0.982 - 0.184i$
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  + (−0.613 + 1.88i)2-s + (−2.37 − 1.72i)4-s + (−0.187 + 0.982i)7-s + (3.10 − 2.25i)8-s + (0.876 + 0.481i)9-s + (0.598 − 0.153i)11-s + (−1.73 − 0.955i)14-s + (1.44 + 4.46i)16-s + (−1.44 + 1.35i)18-s + (−0.0770 + 1.22i)22-s + (−0.5 − 0.363i)23-s + (0.535 + 0.844i)25-s + (2.14 − 2.01i)28-s + (−0.116 + 1.85i)29-s − 5.46·32-s + ⋯
L(s)  = 1  + (−0.613 + 1.88i)2-s + (−2.37 − 1.72i)4-s + (−0.187 + 0.982i)7-s + (3.10 − 2.25i)8-s + (0.876 + 0.481i)9-s + (0.598 − 0.153i)11-s + (−1.73 − 0.955i)14-s + (1.44 + 4.46i)16-s + (−1.44 + 1.35i)18-s + (−0.0770 + 1.22i)22-s + (−0.5 − 0.363i)23-s + (0.535 + 0.844i)25-s + (2.14 − 2.01i)28-s + (−0.116 + 1.85i)29-s − 5.46·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.982 - 0.184i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ -0.982 - 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6703554880\)
\(L(\frac12)\) \(\approx\) \(0.6703554880\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.187 - 0.982i)T \)
151 \( 1 + (-0.728 + 0.684i)T \)
good2 \( 1 + (0.613 - 1.88i)T + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (-0.876 - 0.481i)T^{2} \)
5 \( 1 + (-0.535 - 0.844i)T^{2} \)
11 \( 1 + (-0.598 + 0.153i)T + (0.876 - 0.481i)T^{2} \)
13 \( 1 + (-0.728 + 0.684i)T^{2} \)
17 \( 1 + (0.929 - 0.368i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.116 - 1.85i)T + (-0.992 - 0.125i)T^{2} \)
31 \( 1 + (-0.968 - 0.248i)T^{2} \)
37 \( 1 + (0.574 - 0.227i)T + (0.728 - 0.684i)T^{2} \)
41 \( 1 + (0.425 + 0.904i)T^{2} \)
43 \( 1 + (0.362 + 1.90i)T + (-0.929 + 0.368i)T^{2} \)
47 \( 1 + (0.425 + 0.904i)T^{2} \)
53 \( 1 + (0.683 - 0.825i)T + (-0.187 - 0.982i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.876 + 0.481i)T^{2} \)
67 \( 1 + (-1.27 + 0.702i)T + (0.535 - 0.844i)T^{2} \)
71 \( 1 + (0.328 - 1.72i)T + (-0.929 - 0.368i)T^{2} \)
73 \( 1 + (0.929 + 0.368i)T^{2} \)
79 \( 1 + (-1.96 + 0.248i)T + (0.968 - 0.248i)T^{2} \)
83 \( 1 + (0.637 + 0.770i)T^{2} \)
89 \( 1 + (0.637 + 0.770i)T^{2} \)
97 \( 1 + (-0.535 + 0.844i)T^{2} \)
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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

−10.11597450234589627565757439982, −9.228763427867801622319277953430, −8.797402179887023638548510849771, −7.967018835090194243781889817303, −7.02645532106768140681195550106, −6.56647918697119287189054289506, −5.47830619819875933724052174668, −4.96728226025573297763974823156, −3.77722450719985372012223220975, −1.57042794674705679969418597088, 0.885138265542674246257109914469, 1.99789971438515719443656681889, 3.34598592120961922071458888939, 4.07979535478357307092188801481, 4.69824479538146494159031603449, 6.51192066559207668255555000870, 7.61451382261385189797414697093, 8.275626758945540414946460532945, 9.495990176276708527777901657794, 9.722481316882730479626489148442

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