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Properties

Label 1-6025-6025.1589-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.841 + 0.540i$
Analytic cond. $27.9799$
Root an. cond. $27.9799$
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  + (−0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + 6-s + (0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)18-s + (0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s + 6-s + (0.309 − 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)17-s + (−0.309 − 0.951i)18-s + (0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.841 + 0.540i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.841 + 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05519358852 + 0.1882222047i\)
\(L(\frac12)\) \(\approx\) \(0.05519358852 + 0.1882222047i\)
\(L(1)\) \(\approx\) \(0.6746252537 + 0.03175501498i\)
\(L(1)\) \(\approx\) \(0.6746252537 + 0.03175501498i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.309 + 0.951i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.309 - 0.951i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−17.624421436426124760605889028267, −16.621947982098177159981844254267, −16.22550927269303004710514148266, −15.80103921210096101807271335390, −14.64014142895336219243428694941, −14.1057203757308889622032659380, −13.66091743482910720833464868345, −12.395560316146912462964707363859, −11.99288110326736869938707351127, −11.50768038447736408717636955131, −10.80259618467030299704564143772, −10.262223837104330506078369983030, −9.46059026733385411404291828228, −8.88895383067586474363264087631, −8.521382643706240749338271065022, −7.63073902570346761740186707665, −6.48675570127871377966116310208, −5.58187928910649515805686849, −5.102366011361016153556999665325, −4.41361259947565928964279763739, −3.40731424599446452209606299583, −3.11812700657472307419116510499, −2.14315832106446600089958582305, −1.25750738559443965177957890978, −0.06780281606441046941420234404, 0.99985889361917232306248966972, 1.53022649224090117704751918583, 2.59713348550681349794570727006, 3.7814027509112592256141089326, 4.50443845203415672763640512635, 5.33273767207067667136996919090, 5.93492372760773609921364692395, 6.5617691111846825109816866531, 7.49579779268294308988017836541, 7.72941019968414712362862686686, 8.14190926066739137416080514902, 9.29794718608515625187455735725, 9.9347772656599235369891508339, 10.698756098034726316229905576440, 11.19655575799225908800670775852, 12.33744269392472815812332080189, 12.831474955753590621508305177020, 13.53584970895170641700010398690, 13.973715173303207869907457352382, 14.71222845072631928402830075203, 15.44467433993417805841078736744, 16.03096048583842920071793640934, 16.93236862370756918085838872238, 17.39427163751525119778655224227, 17.86137754786452846943230255422

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