Properties

Label 1-6025-6025.5684-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.191 + 0.981i$
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  + (0.978 + 0.207i)2-s + (−0.669 + 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.978 + 0.207i)11-s + (−0.913 + 0.406i)12-s + (−0.104 + 0.994i)13-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + 17-s + (0.104 − 0.994i)18-s + (−0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (0.978 + 0.207i)2-s + (−0.669 + 0.743i)3-s + (0.913 + 0.406i)4-s + (−0.809 + 0.587i)6-s + (−0.104 − 0.994i)7-s + (0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (0.978 + 0.207i)11-s + (−0.913 + 0.406i)12-s + (−0.104 + 0.994i)13-s + (0.104 − 0.994i)14-s + (0.669 + 0.743i)16-s + 17-s + (0.104 − 0.994i)18-s + (−0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.191 + 0.981i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (5684, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.191 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.919065120 + 2.330351551i\)
\(L(\frac12)\) \(\approx\) \(1.919065120 + 2.330351551i\)
\(L(1)\) \(\approx\) \(1.555536505 + 0.6969021509i\)
\(L(1)\) \(\approx\) \(1.555536505 + 0.6969021509i\)

Euler product

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

−17.33179358141072769495145953123, −17.00089342643168859817889558734, −16.05065532868594620727342205701, −15.56486658853985845434602601639, −14.634516483526519941277725078092, −14.396373146621105775549212041, −13.30532975140283300695664414310, −12.82840844176889967458680133976, −12.39759283763958663473818942343, −11.65424064501921840516194259743, −11.3032142137011133138424820272, −10.45869682336404472084747036830, −9.77313019472679914039404134003, −8.70636625765038804389970036275, −8.03789667920321061548546837966, −7.10830180188969305882943725792, −6.57878017634540768692839190747, −5.88394620004260410219092296864, −5.37877648347969346382504990705, −4.80224138365834447338646737254, −3.72328991111089629470854082649, −2.985903437922434884951424812878, −2.24898273713802333504225130566, −1.49578364285470655529162163157, −0.61475563323444533682575540229, 1.07843174214033384335803377810, 1.78155804521857736379689718184, 3.15574583331509210990902185883, 3.63299009429388718621037436102, 4.39229844114910508104099479984, 4.707779731199650135614343413116, 5.63771299437604927663632510166, 6.4552876624329515536116229309, 6.77849821623821998100538451547, 7.53335256985714297990810144320, 8.54251331480308888766426020546, 9.41348034739420632283542173569, 10.13280763549024064538466783425, 10.71528626080182257245901149106, 11.48743726525888701489841971110, 11.85366622770717506158514206652, 12.62255019605964237231817837984, 13.32490219142203528101363700894, 14.113298288033946455985376480074, 14.76527662700800697793920708840, 14.97879447604744411619308827708, 16.07830926319064778923311343494, 16.63157970577402878348230136676, 16.98029468286914539146280853055, 17.32433962614001489781419695196

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