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Properties

Label 1-6145-6145.1029-r1-0-0
Degree $1$
Conductor $6145$
Sign $0.974 - 0.225i$
Analytic cond. $660.371$
Root an. cond. $660.371$
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  + (−0.302 + 0.953i)2-s + (0.233 + 0.972i)3-s + (−0.817 − 0.576i)4-s + (−0.997 − 0.0715i)6-s + (0.555 + 0.831i)7-s + (0.796 − 0.605i)8-s + (−0.891 + 0.453i)9-s + (0.641 − 0.767i)11-s + (0.369 − 0.929i)12-s + (0.272 + 0.962i)13-s + (−0.960 + 0.277i)14-s + (0.336 + 0.941i)16-s + (−0.902 − 0.430i)17-s + (−0.163 − 0.986i)18-s + (−0.853 + 0.520i)19-s + ⋯
L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.233 + 0.972i)3-s + (−0.817 − 0.576i)4-s + (−0.997 − 0.0715i)6-s + (0.555 + 0.831i)7-s + (0.796 − 0.605i)8-s + (−0.891 + 0.453i)9-s + (0.641 − 0.767i)11-s + (0.369 − 0.929i)12-s + (0.272 + 0.962i)13-s + (−0.960 + 0.277i)14-s + (0.336 + 0.941i)16-s + (−0.902 − 0.430i)17-s + (−0.163 − 0.986i)18-s + (−0.853 + 0.520i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.974 - 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $0.974 - 0.225i$
Analytic conductor: \(660.371\)
Root analytic conductor: \(660.371\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (1:\ ),\ 0.974 - 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.07685547855 + 0.008759032058i\)
\(L(\frac12)\) \(\approx\) \(-0.07685547855 + 0.008759032058i\)
\(L(1)\) \(\approx\) \(0.5303466082 + 0.6805377874i\)
\(L(1)\) \(\approx\) \(0.5303466082 + 0.6805377874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.302 + 0.953i)T \)
3 \( 1 + (0.233 + 0.972i)T \)
7 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (0.641 - 0.767i)T \)
13 \( 1 + (0.272 + 0.962i)T \)
17 \( 1 + (-0.902 - 0.430i)T \)
19 \( 1 + (-0.853 + 0.520i)T \)
23 \( 1 + (-0.122 + 0.992i)T \)
29 \( 1 + (0.350 + 0.936i)T \)
31 \( 1 + (0.900 + 0.435i)T \)
37 \( 1 + (0.444 - 0.895i)T \)
41 \( 1 + (0.823 + 0.567i)T \)
43 \( 1 + (-0.981 - 0.193i)T \)
47 \( 1 + (0.580 + 0.814i)T \)
53 \( 1 + (-0.881 + 0.471i)T \)
59 \( 1 + (0.439 - 0.898i)T \)
61 \( 1 + (0.917 - 0.397i)T \)
67 \( 1 + (-0.959 - 0.282i)T \)
71 \( 1 + (0.848 + 0.529i)T \)
73 \( 1 + (-0.0664 - 0.997i)T \)
79 \( 1 + (0.605 - 0.796i)T \)
83 \( 1 + (-0.843 - 0.537i)T \)
89 \( 1 + (-0.747 - 0.664i)T \)
97 \( 1 + (0.238 - 0.971i)T \)
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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

−17.24949881706744868410129868651, −16.690576070872542612034132232349, −15.304670289560582183044250814311, −14.79796313169997953006693389149, −13.99667446042758103167116648100, −13.383636935368486122378827065369, −12.96160223317656453497818058896, −12.28949457108366389032258196354, −11.59311640415623004184288365559, −11.02763682134180214821261015681, −10.3280401516504909545757121442, −9.704915139435293415171896508700, −8.64008672220877947723315912234, −8.34891068738985783394579902167, −7.685099372102384846545731819661, −6.84676966414148843286404221789, −6.308017954770030542802233212701, −5.11593144126441569384783349225, −4.270062608186546601287898161298, −3.8759787248491851536527323377, −2.64349321619318647163068451361, −2.29145298654094932124638911302, −1.35377084090596449709773621936, −0.77478358291687371495121971670, −0.01352558836470393194990753126, 1.28921048472071760892649737740, 2.11744194923966656640331605128, 3.19488088662040300730403613634, 4.064132783829177994885763149816, 4.607907118038374971129785361167, 5.270817442394909689302168027552, 6.10450140913252243884135054923, 6.473920506993552106218906968, 7.580197569941270955507653152646, 8.37207974007129023940436049059, 8.84027479633183624552740861958, 9.2292105834370966201000860555, 9.92987806781647303855274966926, 10.96647874757450675518586943561, 11.243119684992227327691136686, 12.13398009697003623670379131712, 13.17398898448520776737000981326, 14.04105829121236718467325252512, 14.30113942950993788399075806840, 14.924159491760409581619230181764, 15.76040769260651825107245752772, 15.97543764795583231578033061908, 16.720330096334236576558116357874, 17.35529111184070555388321659516, 17.940149426645376881753656352856

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