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Properties

Label 2-1057-1057.853-c0-0-0
Degree $2$
Conductor $1057$
Sign $-0.0496 - 0.998i$
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  + (−1.56 − 1.13i)2-s + (0.850 + 2.61i)4-s + (−0.929 − 0.368i)7-s + (1.04 − 3.22i)8-s + (0.535 + 0.844i)9-s + (−1.41 + 0.779i)11-s + (1.03 + 1.63i)14-s + (−3.09 + 2.24i)16-s + (0.121 − 1.93i)18-s + (3.10 + 0.392i)22-s + (−0.5 − 1.53i)23-s + (−0.425 + 0.904i)25-s + (0.172 − 2.74i)28-s + (−1.44 − 0.182i)29-s + 4.01·32-s + ⋯
L(s)  = 1  + (−1.56 − 1.13i)2-s + (0.850 + 2.61i)4-s + (−0.929 − 0.368i)7-s + (1.04 − 3.22i)8-s + (0.535 + 0.844i)9-s + (−1.41 + 0.779i)11-s + (1.03 + 1.63i)14-s + (−3.09 + 2.24i)16-s + (0.121 − 1.93i)18-s + (3.10 + 0.392i)22-s + (−0.5 − 1.53i)23-s + (−0.425 + 0.904i)25-s + (0.172 − 2.74i)28-s + (−1.44 − 0.182i)29-s + 4.01·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0496 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0496 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.0496 - 0.998i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ -0.0496 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1309563704\)
\(L(\frac12)\) \(\approx\) \(0.1309563704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.929 + 0.368i)T \)
151 \( 1 + (-0.0627 + 0.998i)T \)
good2 \( 1 + (1.56 + 1.13i)T + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-0.535 - 0.844i)T^{2} \)
5 \( 1 + (0.425 - 0.904i)T^{2} \)
11 \( 1 + (1.41 - 0.779i)T + (0.535 - 0.844i)T^{2} \)
13 \( 1 + (-0.0627 + 0.998i)T^{2} \)
17 \( 1 + (-0.728 + 0.684i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.44 + 0.182i)T + (0.968 + 0.248i)T^{2} \)
31 \( 1 + (-0.876 - 0.481i)T^{2} \)
37 \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \)
41 \( 1 + (0.637 - 0.770i)T^{2} \)
43 \( 1 + (1.62 - 0.645i)T + (0.728 - 0.684i)T^{2} \)
47 \( 1 + (0.637 - 0.770i)T^{2} \)
53 \( 1 + (-0.159 - 0.836i)T + (-0.929 + 0.368i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.535 + 0.844i)T^{2} \)
67 \( 1 + (-0.0672 + 0.106i)T + (-0.425 - 0.904i)T^{2} \)
71 \( 1 + (0.996 + 0.394i)T + (0.728 + 0.684i)T^{2} \)
73 \( 1 + (-0.728 - 0.684i)T^{2} \)
79 \( 1 + (-1.87 + 0.481i)T + (0.876 - 0.481i)T^{2} \)
83 \( 1 + (0.187 - 0.982i)T^{2} \)
89 \( 1 + (0.187 - 0.982i)T^{2} \)
97 \( 1 + (0.425 + 0.904i)T^{2} \)
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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

−10.31174588395456800450535706389, −9.773515034516883319956791199451, −8.881144152247005349468346753471, −7.86939379050595775982399342786, −7.48954511716616022493090541729, −6.57554206136781864362678867338, −4.87817656698159236481842160716, −3.69698320829324384099607507151, −2.66326804551463748162855928580, −1.78582796368061324224836404072, 0.18712968701513081165436282069, 1.97357633652005767016980091786, 3.49131829846114531199517380051, 5.38701674608900839009259871529, 5.82730696926149837681734571317, 6.76539596113935419089018001779, 7.44653520460119934864649566790, 8.269374525860982573033748027149, 9.023005686872221157915815152580, 9.746640219138630411869010927944

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