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Properties

Label 1-2912-2912.731-r0-0-0
Degree $1$
Conductor $2912$
Sign $0.897 - 0.440i$
Analytic cond. $13.5232$
Root an. cond. $13.5232$
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  + (−0.707 + 0.707i)3-s + (0.965 − 0.258i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.707 − 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)25-s + (0.707 + 0.707i)27-s + (−0.258 − 0.965i)29-s + (−0.5 − 0.866i)31-s − 33-s + (0.258 − 0.965i)37-s + (−0.866 − 0.5i)41-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.965 − 0.258i)5-s − i·9-s + (0.707 + 0.707i)11-s + (−0.5 + 0.866i)15-s + (−0.5 + 0.866i)17-s + (0.707 − 0.707i)19-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)25-s + (0.707 + 0.707i)27-s + (−0.258 − 0.965i)29-s + (−0.5 − 0.866i)31-s − 33-s + (0.258 − 0.965i)37-s + (−0.866 − 0.5i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2912 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2912\)    =    \(2^{5} \cdot 7 \cdot 13\)
Sign: $0.897 - 0.440i$
Analytic conductor: \(13.5232\)
Root analytic conductor: \(13.5232\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2912} (731, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2912,\ (0:\ ),\ 0.897 - 0.440i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.550947526 - 0.3597807951i\)
\(L(\frac12)\) \(\approx\) \(1.550947526 - 0.3597807951i\)
\(L(1)\) \(\approx\) \(1.066405820 + 0.05820204224i\)
\(L(1)\) \(\approx\) \(1.066405820 + 0.05820204224i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.965 - 0.258i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.258 - 0.965i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.258 - 0.965i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.866 + 0.5i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−18.93051338942154714991573523826, −18.41169368703032348820402859602, −17.816165606503620679030785975000, −17.17643331996197129385184282324, −16.50789967878783233587344888685, −15.980234223863958592278146406447, −14.71112314717021539021832722750, −14.121230427557011166578571819905, −13.49537951402417908413190971324, −12.923636967180504471552809963524, −12.05246312538259180705717891713, −11.316702769488827825330844016402, −10.84574599637881248495056912141, −9.88785499089957696238421421155, −9.20833914884730104385421668176, −8.38527215632613881970410701727, −7.3359061014357698427554865942, −6.7632642803843939331084654841, −6.12376873635268287867641160473, −5.363603197738046271925988790479, −4.80691789812448097146006491936, −3.39440938535472673119982358653, −2.69843929062347516221159431688, −1.52870784268807504829766836971, −1.12215130284528508990786285185, 0.592072970524013802863807840166, 1.66971040521795218385753574876, 2.52377869681694206532153432554, 3.74585160505633777332853652053, 4.401002091107909193145089233102, 5.19634099005672591661861421952, 5.83838608379276341516427993595, 6.58528222755228406744220025025, 7.23789289333588077345128288558, 8.58103950319076968556734039719, 9.29759779095565907652170869197, 9.6582177287189015888904040906, 10.57217344070472342862968308093, 11.085107461142373946877012964767, 12.01634746387538374638635263679, 12.62674390074457576715391554695, 13.3991692501387170942581792716, 14.19571009150096834405850632145, 15.08975067897424686857490478791, 15.43124162517378821028899105968, 16.50674429093773584096270747721, 17.1364554691698464509919610615, 17.380562132317865238703746456121, 18.16624216010829055299336825157, 18.95050556058249525763747349878

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