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Properties

Label 1-1857-1857.1856-r0-0-0
Degree $1$
Conductor $1857$
Sign $1$
Analytic cond. $8.62387$
Root an. cond. $8.62387$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s + 29-s + 31-s + 32-s + 34-s − 35-s + 37-s − 38-s − 40-s − 41-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s + 22-s − 23-s + 25-s − 26-s + 28-s + 29-s + 31-s + 32-s + 34-s − 35-s + 37-s − 38-s − 40-s − 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1857 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1857 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1857\)    =    \(3 \cdot 619\)
Sign: $1$
Analytic conductor: \(8.62387\)
Root analytic conductor: \(8.62387\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1857} (1856, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1857,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.274856698\)
\(L(\frac12)\) \(\approx\) \(3.274856698\)
\(L(1)\) \(\approx\) \(1.996086374\)
\(L(1)\) \(\approx\) \(1.996086374\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
619 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 - T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 - T \)
73 \( 1 + T \)
79 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.25404800416414842163413034505, −19.45252163744798016500560100832, −19.06383489429621567508556761072, −17.79349597177341980804301462233, −16.93361810545331005112155233651, −16.44414764937697949377899274541, −15.41116803958642639282535829742, −14.81705091030058244471807810692, −14.406021694848799739839704543742, −13.62541666216089088672154398417, −12.3678052981179715932256777139, −12.0516773258085241479167381310, −11.51920876912962200184205673156, −10.624658212761210693220268619330, −9.82943406996214958395261164602, −8.38745119371706346180949032936, −7.97114763467952165743837287394, −7.07705029874303645444205927497, −6.36186223076015470742959076085, −5.28337318502139712582374030496, −4.48666605632926538256213101843, −4.05382347696784224717179211909, −3.040118961746093548518222881966, −2.06816361661260040579275583433, −1.03950904244719509169728871016, 1.03950904244719509169728871016, 2.06816361661260040579275583433, 3.040118961746093548518222881966, 4.05382347696784224717179211909, 4.48666605632926538256213101843, 5.28337318502139712582374030496, 6.36186223076015470742959076085, 7.07705029874303645444205927497, 7.97114763467952165743837287394, 8.38745119371706346180949032936, 9.82943406996214958395261164602, 10.624658212761210693220268619330, 11.51920876912962200184205673156, 12.0516773258085241479167381310, 12.3678052981179715932256777139, 13.62541666216089088672154398417, 14.406021694848799739839704543742, 14.81705091030058244471807810692, 15.41116803958642639282535829742, 16.44414764937697949377899274541, 16.93361810545331005112155233651, 17.79349597177341980804301462233, 19.06383489429621567508556761072, 19.45252163744798016500560100832, 20.25404800416414842163413034505

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