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Properties

Label 1-6013-6013.251-r0-0-0
Degree $1$
Conductor $6013$
Sign $-0.270 + 0.962i$
Analytic cond. $27.9242$
Root an. cond. $27.9242$
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  + (−0.361 + 0.932i)2-s + (−0.0256 + 0.999i)3-s + (−0.738 − 0.674i)4-s + (−0.347 + 0.937i)5-s + (−0.922 − 0.385i)6-s + (0.895 − 0.445i)8-s + (−0.998 − 0.0512i)9-s + (−0.748 − 0.663i)10-s + (−0.984 − 0.174i)11-s + (0.692 − 0.721i)12-s + (−0.841 + 0.540i)13-s + (−0.928 − 0.371i)15-s + (0.0914 + 0.995i)16-s + (−0.856 − 0.515i)17-s + (0.408 − 0.912i)18-s + (−0.327 + 0.945i)19-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)2-s + (−0.0256 + 0.999i)3-s + (−0.738 − 0.674i)4-s + (−0.347 + 0.937i)5-s + (−0.922 − 0.385i)6-s + (0.895 − 0.445i)8-s + (−0.998 − 0.0512i)9-s + (−0.748 − 0.663i)10-s + (−0.984 − 0.174i)11-s + (0.692 − 0.721i)12-s + (−0.841 + 0.540i)13-s + (−0.928 − 0.371i)15-s + (0.0914 + 0.995i)16-s + (−0.856 − 0.515i)17-s + (0.408 − 0.912i)18-s + (−0.327 + 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6013\)    =    \(7 \cdot 859\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(27.9242\)
Root analytic conductor: \(27.9242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6013} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6013,\ (0:\ ),\ -0.270 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1485247333 + 0.1960259050i\)
\(L(\frac12)\) \(\approx\) \(0.1485247333 + 0.1960259050i\)
\(L(1)\) \(\approx\) \(0.3122838842 + 0.3897897845i\)
\(L(1)\) \(\approx\) \(0.3122838842 + 0.3897897845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
859 \( 1 \)
good2 \( 1 + (-0.361 + 0.932i)T \)
3 \( 1 + (-0.0256 + 0.999i)T \)
5 \( 1 + (-0.347 + 0.937i)T \)
11 \( 1 + (-0.984 - 0.174i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.856 - 0.515i)T \)
19 \( 1 + (-0.327 + 0.945i)T \)
23 \( 1 + (-0.948 + 0.316i)T \)
29 \( 1 + (-0.543 - 0.839i)T \)
31 \( 1 + (-0.892 - 0.451i)T \)
37 \( 1 + (0.990 - 0.138i)T \)
41 \( 1 + (0.0695 + 0.997i)T \)
43 \( 1 + (-0.888 - 0.458i)T \)
47 \( 1 + (0.837 + 0.546i)T \)
53 \( 1 + (0.264 - 0.964i)T \)
59 \( 1 + (0.549 + 0.835i)T \)
61 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.972 + 0.232i)T \)
71 \( 1 + (0.938 - 0.344i)T \)
73 \( 1 + (0.221 + 0.975i)T \)
79 \( 1 + (-0.852 - 0.522i)T \)
83 \( 1 + (-0.963 - 0.267i)T \)
89 \( 1 + (-0.967 + 0.253i)T \)
97 \( 1 + (0.743 + 0.668i)T \)
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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

−17.756804118852021114456636569729, −17.040270985411256144599384165, −16.54834115088884289814992901899, −15.601310865060146284339255336879, −14.84861345221845390745005397809, −13.88195971796827474696761128626, −13.216385455227682119654067568472, −12.754238428014900694942201345355, −12.4387707171688308924083092617, −11.673149122689358571997187744867, −10.9795952349181099923783744158, −10.38406168028027659876642621844, −9.44558231792488029953729564515, −8.719444855920477987885737438915, −8.328657635854740141700719046849, −7.52768114281500953041564852685, −7.13458971985908547622128541586, −5.85634159842733842735990141530, −5.13840073071211375047667224260, −4.5424945651550188286765153821, −3.631623618907599532110305280316, −2.639012850633392434080544339610, −2.16918884933659511199173836423, −1.37337694593993948882714115696, −0.352997395773753282439523513920, 0.18266511051836903401498304086, 2.0171339523427837069177783586, 2.67971636709803838948587333471, 3.79238139068904119192970516245, 4.249891309367078680369099049474, 5.04548336065567371159687292079, 5.82307847387614498309342093492, 6.328076468212206913029063589360, 7.30594470650545720237337338664, 7.79547372766608062267533819194, 8.45601658624822911073555992775, 9.34932422321899621697509154322, 9.94997783594397203918137887089, 10.32181308591950251971096597990, 11.211894872917656804705713481142, 11.611904611569797209626237734238, 12.80757515527347206179428187902, 13.68366562021799240569023387255, 14.24810151517909758693351908834, 14.87051678429084744245435894544, 15.29503705526793784361166782024, 15.92566109008414998091814077126, 16.48076795900724108762030067366, 17.05777824214645833766568083272, 17.914884103616028789812411152583

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