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Properties

Label 1-4195-4195.4194-r1-0-0
Degree $1$
Conductor $4195$
Sign $1$
Analytic cond. $450.815$
Root an. cond. $450.815$
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 − 7-s − 8-s + 9-s − 11-s − 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 19-s + 21-s + 22-s − 23-s + 24-s − 26-s − 27-s − 28-s − 29-s − 31-s − 32-s + 33-s − 34-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 11-s − 12-s + 13-s + 14-s + 16-s + 17-s − 18-s + 19-s + 21-s + 22-s − 23-s + 24-s − 26-s − 27-s − 28-s − 29-s − 31-s − 32-s + 33-s − 34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4195 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4195\)    =    \(5 \cdot 839\)
Sign: $1$
Analytic conductor: \(450.815\)
Root analytic conductor: \(450.815\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4195} (4194, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4195,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2594122125\)
\(L(\frac12)\) \(\approx\) \(0.2594122125\)
\(L(1)\) \(\approx\) \(0.3880376601\)
\(L(1)\) \(\approx\) \(0.3880376601\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
839 \( 1 \)
good2 \( 1 - T \)
3 \( 1 - T \)
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

−18.25649176245887669740425991350, −17.690546691997999604605156789545, −16.71862250467170591898007429639, −16.322342850774935062536191076338, −15.85823766326917219395457459922, −15.31881598131136749476496491339, −14.17653064882744141779960045257, −13.08121172914509301092868958203, −12.76565241714874153768249053928, −11.74067707480681186944056851378, −11.43243229862775738896463956039, −10.3463052336293794379206924153, −10.16048035032762944210787618166, −9.46133492179986399978329129372, −8.52635621359988102802890167206, −7.69319012548309952145600848162, −7.11942679832075092888298337551, −6.34295474718166288232516723422, −5.67864106052277287994867076529, −5.203941092707534672133563421094, −3.64174345555824828467430203640, −3.29198203078657115038347748343, −2.00029791621789221797010228885, −1.244448592914520751821415951318, −0.24109381714694480689166144311, 0.24109381714694480689166144311, 1.244448592914520751821415951318, 2.00029791621789221797010228885, 3.29198203078657115038347748343, 3.64174345555824828467430203640, 5.203941092707534672133563421094, 5.67864106052277287994867076529, 6.34295474718166288232516723422, 7.11942679832075092888298337551, 7.69319012548309952145600848162, 8.52635621359988102802890167206, 9.46133492179986399978329129372, 10.16048035032762944210787618166, 10.3463052336293794379206924153, 11.43243229862775738896463956039, 11.74067707480681186944056851378, 12.76565241714874153768249053928, 13.08121172914509301092868958203, 14.17653064882744141779960045257, 15.31881598131136749476496491339, 15.85823766326917219395457459922, 16.322342850774935062536191076338, 16.71862250467170591898007429639, 17.690546691997999604605156789545, 18.25649176245887669740425991350

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