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Properties

Label 1-1137-1137.1136-r0-0-0
Degree $1$
Conductor $1137$
Sign $1$
Analytic cond. $5.28020$
Root an. cond. $5.28020$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1137 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1137\)    =    \(3 \cdot 379\)
Sign: $1$
Analytic conductor: \(5.28020\)
Root analytic conductor: \(5.28020\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1137} (1136, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1137,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.381123394\)
\(L(\frac12)\) \(\approx\) \(2.381123394\)
\(L(1)\) \(\approx\) \(1.654482656\)
\(L(1)\) \(\approx\) \(1.654482656\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
379 \( 1 \)
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

−21.72170515008926437471657270086, −20.25648269509997242140301688220, −19.98673599551782539642203350179, −19.32972891685378211407120105236, −18.508867715864070820992831757975, −17.074771846169276633592429580166, −16.39035491353956784002609582699, −15.919230528844772556170582114528, −14.93990341648669049893803916282, −14.40578668272472399912048148424, −13.53055278995011980767878745383, −12.47636804693897536229747820770, −12.04978071869979681292172495262, −11.50028274006675727584629046602, −10.244073589237651030284886315399, −9.62626723256881043463976592705, −8.31319944501526083283692015840, −7.33426016980424238057786456243, −6.84296089877557630559711796857, −5.82815593146849663013871201893, −4.91813877151357170125031514386, −3.853240093842573386855073908101, −3.43388739870701150989044302234, −2.41856541551362287129879271624, −0.94793428880841682403797747848, 0.94793428880841682403797747848, 2.41856541551362287129879271624, 3.43388739870701150989044302234, 3.853240093842573386855073908101, 4.91813877151357170125031514386, 5.82815593146849663013871201893, 6.84296089877557630559711796857, 7.33426016980424238057786456243, 8.31319944501526083283692015840, 9.62626723256881043463976592705, 10.244073589237651030284886315399, 11.50028274006675727584629046602, 12.04978071869979681292172495262, 12.47636804693897536229747820770, 13.53055278995011980767878745383, 14.40578668272472399912048148424, 14.93990341648669049893803916282, 15.919230528844772556170582114528, 16.39035491353956784002609582699, 17.074771846169276633592429580166, 18.508867715864070820992831757975, 19.32972891685378211407120105236, 19.98673599551782539642203350179, 20.25648269509997242140301688220, 21.72170515008926437471657270086

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