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Properties

Label 1-595-595.594-r1-0-0
Degree $1$
Conductor $595$
Sign $1$
Analytic cond. $63.9416$
Root an. cond. $63.9416$
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 − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 13-s + 16-s − 18-s − 19-s + 22-s + 23-s + 24-s − 26-s − 27-s − 29-s + 31-s − 32-s + 33-s + 36-s + 37-s + 38-s − 39-s + 41-s − 43-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 13-s + 16-s − 18-s − 19-s + 22-s + 23-s + 24-s − 26-s − 27-s − 29-s + 31-s − 32-s + 33-s + 36-s + 37-s + 38-s − 39-s + 41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(595\)    =    \(5 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(63.9416\)
Root analytic conductor: \(63.9416\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{595} (594, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 595,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6962425640\)
\(L(\frac12)\) \(\approx\) \(0.6962425640\)
\(L(1)\) \(\approx\) \(0.5151709684\)
\(L(1)\) \(\approx\) \(0.5151709684\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
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

−23.28613442492412909276907636285, −21.96627951855052983796675310906, −21.05718133971865295247220020140, −20.61573534274718422927869164914, −19.20843609464431664698929584475, −18.650815639886871383962593809920, −17.930726870037727701003923563190, −17.14770131008518939727889578926, −16.41309124839051634132709051498, −15.66091283346163276502025789197, −14.95973196617220197664389489988, −13.28540025604903780027517575641, −12.62330644695811982629292163386, −11.476153419263577763240502147903, −10.88372643706646597804171011369, −10.23998969132813607806057601284, −9.18686137970053743649733094578, −8.17342492969790073140547450349, −7.27846063902029961591110359771, −6.30840887631620980209081799354, −5.614309549512400054084032392507, −4.35066332176245572956405227893, −2.90430138970679852717418859578, −1.62501599285997149546930438665, −0.54201955024844608383065250940, 0.54201955024844608383065250940, 1.62501599285997149546930438665, 2.90430138970679852717418859578, 4.35066332176245572956405227893, 5.614309549512400054084032392507, 6.30840887631620980209081799354, 7.27846063902029961591110359771, 8.17342492969790073140547450349, 9.18686137970053743649733094578, 10.23998969132813607806057601284, 10.88372643706646597804171011369, 11.476153419263577763240502147903, 12.62330644695811982629292163386, 13.28540025604903780027517575641, 14.95973196617220197664389489988, 15.66091283346163276502025789197, 16.41309124839051634132709051498, 17.14770131008518939727889578926, 17.930726870037727701003923563190, 18.650815639886871383962593809920, 19.20843609464431664698929584475, 20.61573534274718422927869164914, 21.05718133971865295247220020140, 21.96627951855052983796675310906, 23.28613442492412909276907636285

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