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Properties

Label 1-2925-2925.644-r0-0-0
Degree $1$
Conductor $2925$
Sign $-0.357 - 0.934i$
Analytic cond. $13.5836$
Root an. cond. $13.5836$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.587 − 0.809i)11-s + (−0.913 − 0.406i)14-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + (0.207 − 0.978i)19-s + (−0.309 + 0.951i)22-s + (0.104 + 0.994i)23-s + (0.207 + 0.978i)28-s + (−0.309 + 0.951i)29-s + (0.743 + 0.669i)31-s + i·32-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.866 − 0.5i)7-s + (0.951 − 0.309i)8-s + (−0.587 − 0.809i)11-s + (−0.913 − 0.406i)14-s + (−0.809 − 0.587i)16-s + (0.978 + 0.207i)17-s + (0.207 − 0.978i)19-s + (−0.309 + 0.951i)22-s + (0.104 + 0.994i)23-s + (0.207 + 0.978i)28-s + (−0.309 + 0.951i)29-s + (0.743 + 0.669i)31-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(13.5836\)
Root analytic conductor: \(13.5836\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2925} (644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2925,\ (0:\ ),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7110491869 - 1.033082354i\)
\(L(\frac12)\) \(\approx\) \(0.7110491869 - 1.033082354i\)
\(L(1)\) \(\approx\) \(0.7733118106 - 0.4077465134i\)
\(L(1)\) \(\approx\) \(0.7733118106 - 0.4077465134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.207 - 0.978i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.743 + 0.669i)T \)
37 \( 1 + (-0.994 - 0.104i)T \)
41 \( 1 + (-0.406 - 0.913i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.743 - 0.669i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.587 - 0.809i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.207 + 0.978i)T \)
71 \( 1 + (0.743 - 0.669i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.207 - 0.978i)T \)
89 \( 1 + (-0.406 + 0.913i)T \)
97 \( 1 + (0.207 + 0.978i)T \)
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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

−18.88979544392273133776902843728, −18.5962894582242265553469898369, −17.96266793307639828576930888215, −17.18233054831425333557533023941, −16.67557653854386327487759259599, −15.764264784541216779289161019702, −15.20225600188274723619923676034, −14.56246451409233909373102238035, −14.041262611342596804348495822899, −13.07468349570135617444511420875, −12.17953343836719087400232564268, −11.51435386997235689456423211958, −10.497747873365045811955121719855, −9.99397654324189804249111212837, −9.27229904802440789827022892258, −8.2296487915167803979228963865, −7.967954524713958967210642998322, −7.20386304642297878863400337774, −6.22087507418824336545700822224, −5.538788075544832669061932415764, −4.857070530830370669082338845024, −4.142267565475553590993149324824, −2.70619769283808239146199821428, −1.870011847021525545471605213467, −0.98981044342162003262402813752, 0.57735722903107320796431195587, 1.40386086571328175789679312487, 2.26982380868013561850653991620, 3.327486247288168573212754488163, 3.76112526630432605306015540944, 5.02916510548565128525823536743, 5.38049631724533794464825465083, 6.93988452117149007974288430234, 7.423046111387944744910005872650, 8.36648003738607701394704596413, 8.6947581716126536156521189631, 9.74684196105696235774035643344, 10.49848995209933298388465890175, 10.938781655718435708050237204297, 11.72266893432856298351482218243, 12.27909647990606092307555409838, 13.39483229656586451574391771685, 13.66888084801903326142212816515, 14.53816682708767031875389947484, 15.56749562171719421051336486982, 16.27509658351928107474231083123, 17.047489878923286348670630830541, 17.610582294254118079903466557835, 18.190618058691957988888255063256, 19.05211510159781399199959528327

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