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Properties

Label 1-5915-5915.1252-r0-0-0
Degree $1$
Conductor $5915$
Sign $-0.587 + 0.809i$
Analytic cond. $27.4691$
Root an. cond. $27.4691$
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  + (0.721 − 0.692i)2-s + (0.600 + 0.799i)3-s + (0.0402 − 0.999i)4-s + (0.987 + 0.160i)6-s + (−0.663 − 0.748i)8-s + (−0.278 + 0.960i)9-s + (−0.278 − 0.960i)11-s + (0.822 − 0.568i)12-s + (−0.996 − 0.0804i)16-s + (−0.979 + 0.200i)17-s + (0.464 + 0.885i)18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.200 − 0.979i)24-s + ⋯
L(s)  = 1  + (0.721 − 0.692i)2-s + (0.600 + 0.799i)3-s + (0.0402 − 0.999i)4-s + (0.987 + 0.160i)6-s + (−0.663 − 0.748i)8-s + (−0.278 + 0.960i)9-s + (−0.278 − 0.960i)11-s + (0.822 − 0.568i)12-s + (−0.996 − 0.0804i)16-s + (−0.979 + 0.200i)17-s + (0.464 + 0.885i)18-s + (0.5 − 0.866i)19-s + (−0.866 − 0.5i)22-s + (−0.866 + 0.5i)23-s + (0.200 − 0.979i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.587 + 0.809i$
Analytic conductor: \(27.4691\)
Root analytic conductor: \(27.4691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (1252, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (0:\ ),\ -0.587 + 0.809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1091969522 + 0.2142086204i\)
\(L(\frac12)\) \(\approx\) \(0.1091969522 + 0.2142086204i\)
\(L(1)\) \(\approx\) \(1.322974475 - 0.3261346390i\)
\(L(1)\) \(\approx\) \(1.322974475 - 0.3261346390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.721 - 0.692i)T \)
3 \( 1 + (0.600 + 0.799i)T \)
11 \( 1 + (-0.278 - 0.960i)T \)
17 \( 1 + (-0.979 + 0.200i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.692 - 0.721i)T \)
31 \( 1 + (-0.354 - 0.935i)T \)
37 \( 1 + (0.774 + 0.632i)T \)
41 \( 1 + (0.799 - 0.600i)T \)
43 \( 1 + (-0.774 + 0.632i)T \)
47 \( 1 + (-0.464 + 0.885i)T \)
53 \( 1 + (0.663 + 0.748i)T \)
59 \( 1 + (0.996 - 0.0804i)T \)
61 \( 1 + (-0.948 + 0.316i)T \)
67 \( 1 + (-0.999 + 0.0402i)T \)
71 \( 1 + (0.919 + 0.391i)T \)
73 \( 1 + (-0.239 + 0.970i)T \)
79 \( 1 + (-0.885 - 0.464i)T \)
83 \( 1 + (-0.992 - 0.120i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.903 + 0.428i)T \)
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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

−17.738052046077124687721113769701, −16.66504645692964523385515682909, −16.18325398299908330040353869135, −15.31610048025248591067706144353, −14.80147247180669935372793547726, −14.276719096879449377733630322180, −13.61093979941813219110411146915, −12.97203453121105842106713209250, −12.47480578840889153726526313027, −11.92167797187180669351599305005, −11.14108993718872965282729330652, −10.06926823597002746810975984282, −9.27371535758353759827565645116, −8.56310574849148239036643701925, −7.973892276726864361076503606, −7.22962353812842832842270930488, −6.88255859265823932938539625376, −6.04988958619087532436408082363, −5.359948187292191566261285207487, −4.47438776648482079648811760195, −3.79478641414018094639914578008, −3.02938896473531569704256092276, −2.203526140976048045460340327067, −1.64365153962482909073758778584, −0.037439353681108350280275806319, 1.22766957600816624144823665223, 2.3293329059993286229007496025, 2.70782484473927896268987348121, 3.582800383007942514016460984272, 4.16366208900590667992541632171, 4.76987504793060459570079002285, 5.67153141214201040192209515235, 6.078473634950158319335353912093, 7.2142502973475484772744942143, 8.06501457408642922049381247077, 8.82847146077380894110669199276, 9.510322087226965100665929865631, 9.95306020346059920579006776469, 10.98119673456663432807857188398, 11.153518249601244306640952543489, 11.85908594301726141965097016820, 13.01384347021609435752178984626, 13.44653706284934438683327531552, 13.84429217872412668318346110305, 14.72062011062366646653284023326, 15.20795506621818740432837330421, 15.858936703940263179459169377210, 16.32433241251653531892787284490, 17.30026398438919177630185466361, 18.16148009764072288278257915732

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