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Properties

Label 1-6145-6145.1104-r0-0-0
Degree $1$
Conductor $6145$
Sign $-0.539 - 0.841i$
Analytic cond. $28.5372$
Root an. cond. $28.5372$
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  + (−0.940 + 0.340i)2-s + (0.774 + 0.633i)3-s + (0.767 − 0.641i)4-s + (−0.943 − 0.331i)6-s + (−0.989 − 0.142i)7-s + (−0.502 + 0.864i)8-s + (0.198 + 0.980i)9-s + (0.355 − 0.934i)11-s + (0.999 − 0.0102i)12-s + (−0.588 − 0.808i)13-s + (0.979 − 0.203i)14-s + (0.178 − 0.984i)16-s + (−0.683 + 0.730i)17-s + (−0.520 − 0.853i)18-s + (−0.993 + 0.112i)19-s + ⋯
L(s)  = 1  + (−0.940 + 0.340i)2-s + (0.774 + 0.633i)3-s + (0.767 − 0.641i)4-s + (−0.943 − 0.331i)6-s + (−0.989 − 0.142i)7-s + (−0.502 + 0.864i)8-s + (0.198 + 0.980i)9-s + (0.355 − 0.934i)11-s + (0.999 − 0.0102i)12-s + (−0.588 − 0.808i)13-s + (0.979 − 0.203i)14-s + (0.178 − 0.984i)16-s + (−0.683 + 0.730i)17-s + (−0.520 − 0.853i)18-s + (−0.993 + 0.112i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6145\)    =    \(5 \cdot 1229\)
Sign: $-0.539 - 0.841i$
Analytic conductor: \(28.5372\)
Root analytic conductor: \(28.5372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6145} (1104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6145,\ (0:\ ),\ -0.539 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05288116419 - 0.09674369981i\)
\(L(\frac12)\) \(\approx\) \(0.05288116419 - 0.09674369981i\)
\(L(1)\) \(\approx\) \(0.6611194678 + 0.1827589546i\)
\(L(1)\) \(\approx\) \(0.6611194678 + 0.1827589546i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
1229 \( 1 \)
good2 \( 1 + (-0.940 + 0.340i)T \)
3 \( 1 + (0.774 + 0.633i)T \)
7 \( 1 + (-0.989 - 0.142i)T \)
11 \( 1 + (0.355 - 0.934i)T \)
13 \( 1 + (-0.588 - 0.808i)T \)
17 \( 1 + (-0.683 + 0.730i)T \)
19 \( 1 + (-0.993 + 0.112i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (0.117 + 0.993i)T \)
31 \( 1 + (0.946 + 0.321i)T \)
37 \( 1 + (0.866 - 0.498i)T \)
41 \( 1 + (-0.988 - 0.152i)T \)
43 \( 1 + (-0.909 + 0.416i)T \)
47 \( 1 + (-0.900 + 0.435i)T \)
53 \( 1 + (0.997 - 0.0715i)T \)
59 \( 1 + (0.0970 + 0.995i)T \)
61 \( 1 + (-0.834 + 0.550i)T \)
67 \( 1 + (-0.218 - 0.975i)T \)
71 \( 1 + (0.740 - 0.671i)T \)
73 \( 1 + (0.690 + 0.723i)T \)
79 \( 1 + (0.502 + 0.864i)T \)
83 \( 1 + (-0.0153 - 0.999i)T \)
89 \( 1 + (0.817 - 0.576i)T \)
97 \( 1 + (0.895 + 0.444i)T \)
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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

−18.091408087445982249934832471451, −17.11916342653963221453536991415, −17.01228226537713107617755835564, −15.92903862872696981034591205300, −15.19427439196312553873241275384, −14.963737182733960220978593702359, −13.73321763134357756166678878965, −13.21244314425391919861647229900, −12.64256268867802919194885715152, −11.767143414239014786538649210801, −11.64139673538674359436173518721, −10.316426588397289141279168124526, −9.668000271268671947837134730555, −9.3724634399369462323097008793, −8.69183145148696682517699786098, −7.95187265515173758377857070326, −7.17371515429049045432725146615, −6.62954920933827250251594760615, −6.37156508488839235061812687217, −4.80728870382181534973830562982, −3.96511491541130837939704262178, −3.22108444442691801080774487319, −2.31517312162919768531384813039, −2.12772846126008531451951509506, −1.02530537589459859249077294664, 0.03680954020043820989759971658, 1.15338376134724464123272761627, 2.23896130899841457246610756364, 2.91517500650480000118387297564, 3.476052068772127077007677805058, 4.48803711592149010747289543373, 5.301278905893214993111953096410, 6.26673462416911944816556672785, 6.660679315933685335148750438852, 7.571553921557013311472009773483, 8.39898054335481521274755140636, 8.70397844241537160626567100037, 9.37619399187549365840686794766, 10.1517288746785135992508287870, 10.56156790933312171580495303585, 11.08642199994362452175420400903, 12.19620755045484949953685156439, 12.98367420426107158351411182961, 13.58672944925208007376632195705, 14.496791076881592423748167032257, 15.04128198189322319148890565370, 15.448256026326035614050703997789, 16.34948507812857494299043044049, 16.64991005013577086074756323146, 17.22575739657436430368917427549

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