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Properties

Label 1-237-237.236-r0-0-0
Degree $1$
Conductor $237$
Sign $1$
Analytic cond. $1.10062$
Root an. cond. $1.10062$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 237 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(237\)    =    \(3 \cdot 79\)
Sign: $1$
Analytic conductor: \(1.10062\)
Root analytic conductor: \(1.10062\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{237} (236, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 237,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5437225553\)
\(L(\frac12)\) \(\approx\) \(0.5437225553\)
\(L(1)\) \(\approx\) \(0.5642990488\)
\(L(1)\) \(\approx\) \(0.5642990488\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
79 \( 1 \)
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 \( 1 + T \)
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

−26.228483903842244351463549684335, −25.696681553346080431861429682629, −24.50817868640428240532606971867, −23.49141127266889308549994644742, −22.80788094104506386683739727263, −21.311782642079020925396546223234, −20.3909112757487933294686148424, −19.57912922300236549694298570770, −18.73753602260964681423952525666, −18.10408755453157156174789323688, −16.65677776981729349694623084453, −15.84410981400718538187860806256, −15.55539810075170448117558703200, −13.89815352259521864991385618524, −12.510949505834764074712760744965, −11.76926354839747336840673723066, −10.581216472342380051806003112321, −9.8361855640877769696132784310, −8.527746936949063057607789870508, −7.79162830510761824576641994744, −6.758833178613469753776362185758, −5.58211481877082461286971589792, −3.68374247176938053157890360759, −2.787618397182406795517156850177, −0.84239477384849489650252279285, 0.84239477384849489650252279285, 2.787618397182406795517156850177, 3.68374247176938053157890360759, 5.58211481877082461286971589792, 6.758833178613469753776362185758, 7.79162830510761824576641994744, 8.527746936949063057607789870508, 9.8361855640877769696132784310, 10.581216472342380051806003112321, 11.76926354839747336840673723066, 12.510949505834764074712760744965, 13.89815352259521864991385618524, 15.55539810075170448117558703200, 15.84410981400718538187860806256, 16.65677776981729349694623084453, 18.10408755453157156174789323688, 18.73753602260964681423952525666, 19.57912922300236549694298570770, 20.3909112757487933294686148424, 21.311782642079020925396546223234, 22.80788094104506386683739727263, 23.49141127266889308549994644742, 24.50817868640428240532606971867, 25.696681553346080431861429682629, 26.228483903842244351463549684335

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