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Properties

Label 1-6025-6025.231-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.820 + 0.571i$
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  + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)6-s + (0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (0.913 + 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)6-s + (0.669 − 0.743i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.913 + 0.406i)11-s + (−0.5 − 0.866i)12-s + (−0.5 + 0.866i)13-s + (0.913 − 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.809 + 0.587i)17-s + (0.669 + 0.743i)18-s + (0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.843651248 + 0.8922623540i\)
\(L(\frac12)\) \(\approx\) \(2.843651248 + 0.8922623540i\)
\(L(1)\) \(\approx\) \(1.581641135 + 0.3585689589i\)
\(L(1)\) \(\approx\) \(1.581641135 + 0.3585689589i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.913 + 0.406i)T \)
3 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (0.669 - 0.743i)T \)
11 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.669 - 0.743i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.913 + 0.406i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−17.63681115457232454756117819490, −16.992237765006207880459439200, −16.15205802283833021322475557324, −15.61993226506466644373886848233, −14.95697166069261001846442568994, −14.5305444683585420168186961965, −13.4861955822153887387136265084, −13.08886824326642991662730064510, −12.05437078193032741520961805587, −11.76637029448877815749764503005, −11.35809810798433747724628413852, −10.63417221265059460938897805317, −9.67493783547115451169552164592, −9.44662775857866683001099673662, −8.12295948906517814508010998837, −7.42096202248239648971524096217, −6.48115580795184940574646504815, −5.94238397722725772394452350081, −5.40738670777586325687953106490, −4.684610867919145170069872171576, −4.17907237447424259620385769397, −3.19160309936868957219282725594, −2.478645095619058054622826505620, −1.43667363978826624348523340111, −0.85221088736406642397978400695, 0.811404244094640360100798723181, 1.78436982906491391381164764446, 2.38113800627750622050672737948, 3.80082715582396331398676361032, 4.21230566607392344812865853209, 4.890442670591665736787845256, 5.3656370473438814044832523622, 6.47946655560236465931060273459, 6.91752437939895531108418613784, 7.17729050886766468639562729242, 8.2281064708870151083600766410, 8.99130157939372451301653143385, 10.03540583541309154293177868009, 10.81671902214310320526570127781, 11.372964707899205707670671524946, 11.80940258538806327808858116441, 12.59848145235066300242569784875, 13.06152004793746226891688494296, 14.0405229948009023176949627114, 14.294173545956768764626765918300, 15.12763935206494149734875036534, 15.86034802655905878755063760752, 16.527062304140038106914384688511, 17.10257511695369341683713711096, 17.480243862462426042535455078006

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