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Properties

Label 1-4645-4645.4644-r0-0-0
Degree $1$
Conductor $4645$
Sign $1$
Analytic cond. $21.5712$
Root an. cond. $21.5712$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 4645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4645\)    =    \(5 \cdot 929\)
Sign: $1$
Analytic conductor: \(21.5712\)
Root analytic conductor: \(21.5712\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4645} (4644, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 4645,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.604072542\)
\(L(\frac12)\) \(\approx\) \(2.604072542\)
\(L(1)\) \(\approx\) \(1.366801319\)
\(L(1)\) \(\approx\) \(1.366801319\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
929 \( 1 \)
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

−18.35905691611642904731251161202, −17.64899556985280088955343102919, −16.97481980599414415112242962100, −16.02277265574388737008043187628, −15.75602409232963996743461755998, −14.78613817156076597779147250227, −14.201316976842190105821115938085, −13.8828147065899779751777280062, −12.62213880030306884197751644242, −11.97002764653920914435360824168, −11.34910953424927913246598557086, −10.558285978034435938007862393631, −9.83130489526452675203082179696, −9.25337565399859354188300699232, −8.479908073588048471757779724410, −8.119971663722176031129548630171, −7.41818692643997469790499900644, −6.70471658197722465099492251979, −5.84820923505274232388815112306, −4.89740969901297682799631131429, −3.70688611199162038865771137134, −3.39163379327363633678629944473, −2.24377582939230978562030225106, −1.48571124098308941510436819518, −1.09153637070162642212145484445, 1.09153637070162642212145484445, 1.48571124098308941510436819518, 2.24377582939230978562030225106, 3.39163379327363633678629944473, 3.70688611199162038865771137134, 4.89740969901297682799631131429, 5.84820923505274232388815112306, 6.70471658197722465099492251979, 7.41818692643997469790499900644, 8.119971663722176031129548630171, 8.479908073588048471757779724410, 9.25337565399859354188300699232, 9.83130489526452675203082179696, 10.558285978034435938007862393631, 11.34910953424927913246598557086, 11.97002764653920914435360824168, 12.62213880030306884197751644242, 13.8828147065899779751777280062, 14.201316976842190105821115938085, 14.78613817156076597779147250227, 15.75602409232963996743461755998, 16.02277265574388737008043187628, 16.97481980599414415112242962100, 17.64899556985280088955343102919, 18.35905691611642904731251161202

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