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Properties

Label 1-2631-2631.2630-r1-0-0
Degree $1$
Conductor $2631$
Sign $1$
Analytic cond. $282.740$
Root an. cond. $282.740$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2631 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2631 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2631\)    =    \(3 \cdot 877\)
Sign: $1$
Analytic conductor: \(282.740\)
Root analytic conductor: \(282.740\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2631} (2630, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 2631,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(8.244578843\)
\(L(\frac12)\) \(\approx\) \(8.244578843\)
\(L(1)\) \(\approx\) \(2.939887326\)
\(L(1)\) \(\approx\) \(2.939887326\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
877 \( 1 \)
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

−19.2802907326854190994073418419, −18.51231516252559102394210667839, −17.39981451365559581261928508915, −17.07878310770552320871770271239, −16.487483535108799000725280388506, −15.22600712548593214742133726947, −14.7481562234155679044627140406, −14.12489087771555367679704677668, −13.75547832714032707467906997709, −12.63497691308622735944196427312, −12.18236328418853637567779375025, −11.42732435832969081086948493985, −10.60478504600775233441536093662, −9.93030938621463771140939324210, −9.100104034427587685487563962045, −7.99380857700826262311351172929, −7.37630103696490700050232322886, −6.32223828893433641416868190754, −5.90866235533756428079710390831, −4.95965411164139969712179886944, −4.43789576739156699889478577285, −3.489004060870498332874196420369, −2.356574372417942603225712826384, −1.88253493613696838629445215081, −0.98585460842313739189283215371, 0.98585460842313739189283215371, 1.88253493613696838629445215081, 2.356574372417942603225712826384, 3.489004060870498332874196420369, 4.43789576739156699889478577285, 4.95965411164139969712179886944, 5.90866235533756428079710390831, 6.32223828893433641416868190754, 7.37630103696490700050232322886, 7.99380857700826262311351172929, 9.100104034427587685487563962045, 9.93030938621463771140939324210, 10.60478504600775233441536093662, 11.42732435832969081086948493985, 12.18236328418853637567779375025, 12.63497691308622735944196427312, 13.75547832714032707467906997709, 14.12489087771555367679704677668, 14.7481562234155679044627140406, 15.22600712548593214742133726947, 16.487483535108799000725280388506, 17.07878310770552320871770271239, 17.39981451365559581261928508915, 18.51231516252559102394210667839, 19.2802907326854190994073418419

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