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Properties

Label 2-1057-1057.601-c0-0-0
Degree $2$
Conductor $1057$
Sign $-0.205 + 0.978i$
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  + (1.50 − 1.09i)2-s + (0.759 − 2.33i)4-s + (0.535 − 0.844i)7-s + (−0.837 − 2.57i)8-s + (0.0627 + 0.998i)9-s + (−1.17 + 1.10i)11-s + (−0.116 − 1.85i)14-s + (−2.09 − 1.51i)16-s + (1.18 + 1.43i)18-s + (−0.563 + 2.95i)22-s + (−0.5 + 1.53i)23-s + (−0.992 + 0.125i)25-s + (−1.56 − 1.89i)28-s + (0.159 − 0.836i)29-s − 2.09·32-s + ⋯
L(s)  = 1  + (1.50 − 1.09i)2-s + (0.759 − 2.33i)4-s + (0.535 − 0.844i)7-s + (−0.837 − 2.57i)8-s + (0.0627 + 0.998i)9-s + (−1.17 + 1.10i)11-s + (−0.116 − 1.85i)14-s + (−2.09 − 1.51i)16-s + (1.18 + 1.43i)18-s + (−0.563 + 2.95i)22-s + (−0.5 + 1.53i)23-s + (−0.992 + 0.125i)25-s + (−1.56 − 1.89i)28-s + (0.159 − 0.836i)29-s − 2.09·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1057 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1057\)    =    \(7 \cdot 151\)
Sign: $-0.205 + 0.978i$
Analytic conductor: \(0.527511\)
Root analytic conductor: \(0.726300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1057} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1057,\ (\ :0),\ -0.205 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.222064649\)
\(L(\frac12)\) \(\approx\) \(2.222064649\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.535 + 0.844i)T \)
151 \( 1 + (0.637 + 0.770i)T \)
good2 \( 1 + (-1.50 + 1.09i)T + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.0627 - 0.998i)T^{2} \)
5 \( 1 + (0.992 - 0.125i)T^{2} \)
11 \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \)
13 \( 1 + (0.637 + 0.770i)T^{2} \)
17 \( 1 + (0.425 - 0.904i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.159 + 0.836i)T + (-0.929 - 0.368i)T^{2} \)
31 \( 1 + (-0.728 - 0.684i)T^{2} \)
37 \( 1 + (-0.688 + 1.46i)T + (-0.637 - 0.770i)T^{2} \)
41 \( 1 + (-0.968 - 0.248i)T^{2} \)
43 \( 1 + (-0.781 - 1.23i)T + (-0.425 + 0.904i)T^{2} \)
47 \( 1 + (-0.968 - 0.248i)T^{2} \)
53 \( 1 + (1.73 + 0.955i)T + (0.535 + 0.844i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.0627 + 0.998i)T^{2} \)
67 \( 1 + (0.0800 - 1.27i)T + (-0.992 - 0.125i)T^{2} \)
71 \( 1 + (-0.0672 + 0.106i)T + (-0.425 - 0.904i)T^{2} \)
73 \( 1 + (0.425 + 0.904i)T^{2} \)
79 \( 1 + (-1.72 + 0.684i)T + (0.728 - 0.684i)T^{2} \)
83 \( 1 + (-0.876 + 0.481i)T^{2} \)
89 \( 1 + (-0.876 + 0.481i)T^{2} \)
97 \( 1 + (0.992 + 0.125i)T^{2} \)
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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

−10.05533549978621752223803355882, −9.746877731865748876766404923904, −7.81996476042799135610600377175, −7.45962185230799684815041931662, −6.02395513276639289247349109535, −5.17124207714423886534837290695, −4.55965634468113125556818323716, −3.76370601957843325522483167188, −2.44687767072302275889553648461, −1.71571792009562035541334100186, 2.54098747624550510616533870049, 3.36128044586726685779322316279, 4.46065142423199238740395490259, 5.29255053382430649095360271986, 6.04127707134388917812894944858, 6.56065322878815178886798056213, 7.84162788016744804560302025124, 8.258315693462145690919792073696, 9.152040358141508502686623910374, 10.58016995156318064131038970418

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