↑

Properties

Label 1-5915-5915.1003-r0-0-0
Degree $1$
Conductor $5915$
Sign $-0.778 + 0.628i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.885 − 0.464i)2-s + (−0.960 + 0.278i)3-s + (0.568 + 0.822i)4-s + (0.979 + 0.200i)6-s + (−0.120 − 0.992i)8-s + (0.845 − 0.534i)9-s + (0.534 − 0.845i)11-s + (−0.774 − 0.632i)12-s + (−0.354 + 0.935i)16-s + (−0.992 + 0.120i)17-s + (−0.996 + 0.0804i)18-s + (−0.866 − 0.5i)19-s + (−0.866 + 0.5i)22-s − i·23-s + (0.391 + 0.919i)24-s + ⋯
L(s)  = 1  + (−0.885 − 0.464i)2-s + (−0.960 + 0.278i)3-s + (0.568 + 0.822i)4-s + (0.979 + 0.200i)6-s + (−0.120 − 0.992i)8-s + (0.845 − 0.534i)9-s + (0.534 − 0.845i)11-s + (−0.774 − 0.632i)12-s + (−0.354 + 0.935i)16-s + (−0.992 + 0.120i)17-s + (−0.996 + 0.0804i)18-s + (−0.866 − 0.5i)19-s + (−0.866 + 0.5i)22-s − i·23-s + (0.391 + 0.919i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5915 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5915\)    =    \(5 \cdot 7 \cdot 13^{2}\)
Sign: $-0.778 + 0.628i$
Analytic conductor: \(27.4691\)
Root analytic conductor: \(27.4691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5915} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5915,\ (0:\ ),\ -0.778 + 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07989687840 - 0.2261564119i\)
\(L(\frac12)\) \(\approx\) \(-0.07989687840 - 0.2261564119i\)
\(L(1)\) \(\approx\) \(0.4425720593 - 0.1585213362i\)
\(L(1)\) \(\approx\) \(0.4425720593 - 0.1585213362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.885 - 0.464i)T \)
3 \( 1 + (-0.960 + 0.278i)T \)
11 \( 1 + (0.534 - 0.845i)T \)
17 \( 1 + (-0.992 + 0.120i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.845 - 0.534i)T \)
31 \( 1 + (-0.316 - 0.948i)T \)
37 \( 1 + (-0.748 - 0.663i)T \)
41 \( 1 + (-0.721 - 0.692i)T \)
43 \( 1 + (0.316 - 0.948i)T \)
47 \( 1 + (-0.996 - 0.0804i)T \)
53 \( 1 + (0.391 - 0.919i)T \)
59 \( 1 + (-0.935 + 0.354i)T \)
61 \( 1 + (0.799 + 0.600i)T \)
67 \( 1 + (0.996 + 0.0804i)T \)
71 \( 1 + (0.960 - 0.278i)T \)
73 \( 1 + (0.0402 - 0.999i)T \)
79 \( 1 + (0.996 + 0.0804i)T \)
83 \( 1 + (-0.970 + 0.239i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.987 + 0.160i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.01911604340607664972641103722, −17.39848086100672699234321869246, −17.08608358672313357955032031940, −16.28776351962448766566794797863, −15.68214914307085777611602695662, −15.151136172640890198989959566359, −14.34221304272178738803039800184, −13.59021954224718849345316118006, −12.64845514668596887689344739192, −12.14110048714732885332356344464, −11.30773621938276740884088773478, −10.90464141387710556179650717759, −10.05923800800098134403411859696, −9.63190745494382732648140591934, −8.72703246309864983706532498010, −8.05634481261782226564383782580, −7.25139222031299458461920379027, −6.538209297751140702629155823532, −6.42392160389990802511243162964, −5.22765803916386948960878604070, −4.851261918071748641847816209824, −3.88865461043973430073250643685, −2.59213488204803876396593732701, −1.64011844178862501213156265296, −1.283436914012207139962002699576, 0.13429013655440509350030251966, 0.72029198324136033915411624732, 1.83445188550574417493495235937, 2.51266104386469314959224720555, 3.66839613258360405829758941594, 4.12670953777162180625274084726, 5.009388496630518819083880751861, 6.045466739037198145639191294742, 6.59281299337377918423300865056, 7.06278766986960356537784755565, 8.19773559637810812279236814339, 8.754249478335344488113355111099, 9.34607755010659251322159708772, 10.23327333394972342952968676997, 10.708339596756178602602203266966, 11.277258893982326374235634660, 11.80771441984372186892059499513, 12.53476626335253643378692630874, 13.120136877873183465094806200245, 13.943497929611736337108391203677, 15.081538278359119238630593450675, 15.56613115922006284538612517964, 16.30407373169094624010917651625, 16.83493327958096692228438358737, 17.34026027475251539599208444553

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