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Properties

Label 1-5915-5915.1087-r0-0-0
Degree $1$
Conductor $5915$
Sign $-0.0954 - 0.995i$
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  + (0.0402 + 0.999i)2-s + (−0.721 − 0.692i)3-s + (−0.996 + 0.0804i)4-s + (0.663 − 0.748i)6-s + (−0.120 − 0.992i)8-s + (0.0402 + 0.999i)9-s + (0.999 + 0.0402i)11-s + (0.774 + 0.632i)12-s + (0.987 − 0.160i)16-s + (−0.391 + 0.919i)17-s + (−0.996 + 0.0804i)18-s + (−0.866 + 0.5i)19-s + i·22-s + (0.866 − 0.5i)23-s + (−0.600 + 0.799i)24-s + ⋯
L(s)  = 1  + (0.0402 + 0.999i)2-s + (−0.721 − 0.692i)3-s + (−0.996 + 0.0804i)4-s + (0.663 − 0.748i)6-s + (−0.120 − 0.992i)8-s + (0.0402 + 0.999i)9-s + (0.999 + 0.0402i)11-s + (0.774 + 0.632i)12-s + (0.987 − 0.160i)16-s + (−0.391 + 0.919i)17-s + (−0.996 + 0.0804i)18-s + (−0.866 + 0.5i)19-s + i·22-s + (0.866 − 0.5i)23-s + (−0.600 + 0.799i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0954 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.0954 - 0.995i$
Analytic conductor: \(27.4691\)
Root analytic conductor: \(27.4691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (1087, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (0:\ ),\ -0.0954 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1531178792 - 0.1685099345i\)
\(L(\frac12)\) \(\approx\) \(0.1531178792 - 0.1685099345i\)
\(L(1)\) \(\approx\) \(0.6499525252 + 0.2153106076i\)
\(L(1)\) \(\approx\) \(0.6499525252 + 0.2153106076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.0402 + 0.999i)T \)
3 \( 1 + (-0.721 - 0.692i)T \)
11 \( 1 + (0.999 + 0.0402i)T \)
17 \( 1 + (-0.391 + 0.919i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (0.316 + 0.948i)T \)
37 \( 1 + (-0.200 + 0.979i)T \)
41 \( 1 + (0.239 - 0.970i)T \)
43 \( 1 + (-0.663 - 0.748i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
53 \( 1 + (-0.391 + 0.919i)T \)
59 \( 1 + (-0.160 + 0.987i)T \)
61 \( 1 + (-0.919 + 0.391i)T \)
67 \( 1 + (-0.428 - 0.903i)T \)
71 \( 1 + (0.239 - 0.970i)T \)
73 \( 1 + (0.0402 - 0.999i)T \)
79 \( 1 + (0.996 + 0.0804i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 + (0.866 - 0.5i)T \)
97 \( 1 + (0.354 - 0.935i)T \)
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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

−17.76147189923475591239845464287, −17.43960354470074340308936711414, −16.71597950121810351639395840706, −16.12795719368168620797007193327, −15.05510714885067819277658241475, −14.7610351977756435210007906656, −13.91588976862064888058062778227, −13.02362443166496256257473890219, −12.64352539732817950949926515287, −11.52972037788406826561932466778, −11.438763255274264448467377089301, −10.864599942377727598254108980661, −9.92489654184580679594791440852, −9.36745251220910220863081147518, −9.030302234082195551607880356567, −8.07560107597476052836857818622, −6.949015037761240238925222548150, −6.31931845421668434706078355729, −5.41553193475885790856852661919, −4.801893546407387443789064737115, −4.19076046924676603982749555135, −3.51556918762535418049685786414, −2.77108563242062379069641044663, −1.74712331932423496619556628539, −0.90202864760023258880869286603, 0.07941406753766376076476204169, 1.253283258204310545564087068843, 1.88320974572494469637921826826, 3.2104194206065136029543653286, 4.12484862648786095301577787003, 4.714749372127503560805091982100, 5.51158707347618570624976638844, 6.36256419144324480592153738133, 6.506058514345689643920606597882, 7.31685863356916325977424333635, 8.06449782104905802255035200729, 8.69804000242600784092451376022, 9.317580251595295825250160895, 10.43323957942777071617166543739, 10.775347379436989120942987248120, 11.95940160208719943906186380691, 12.288887012637395575176724466008, 13.10885110552803582881181681312, 13.56809825527229997730901019542, 14.37189068307719593788999412254, 14.99866520855351547705992607401, 15.562196820987919501554222093224, 16.62155569345845939301664341304, 16.90282795061300619741778859095, 17.31517728064261712137947594565

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