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Properties

Label 1-6025-6025.804-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.125 - 0.992i$
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 + (0.104 − 0.994i)3-s + (0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.913 + 0.406i)11-s + (−0.669 − 0.743i)12-s + (−0.978 + 0.207i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + 17-s + (0.978 − 0.207i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.104 − 0.994i)3-s + (0.669 − 0.743i)4-s + (0.309 + 0.951i)6-s + (−0.978 − 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.978 − 0.207i)9-s + (−0.913 + 0.406i)11-s + (−0.669 − 0.743i)12-s + (−0.978 + 0.207i)13-s + (0.978 − 0.207i)14-s + (−0.104 − 0.994i)16-s + 17-s + (0.978 − 0.207i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.125 - 0.992i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (804, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.125 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3186512365 - 0.3616272927i\)
\(L(\frac12)\) \(\approx\) \(0.3186512365 - 0.3616272927i\)
\(L(1)\) \(\approx\) \(0.5210327450 - 0.1087279633i\)
\(L(1)\) \(\approx\) \(0.5210327450 - 0.1087279633i\)

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 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.978 - 0.207i)T \)
11 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + T \)
19 \( 1 + (-0.669 - 0.743i)T \)
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 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
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 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
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

−17.83030277127911173187290388854, −17.069775394992415570923423849645, −16.59908309371610121941193618979, −16.11409716756677565071329601742, −15.44566562351432736632707550087, −14.88377048031021862903296251997, −14.069931641066388682954863709741, −12.93237354779379784385604225999, −12.66419731052069584727332029513, −11.74637443274238266513181633865, −11.109117442193475711898307688151, −10.2742605148167815480680236139, −9.97768230352441309030424433551, −9.49614884441970411216262202215, −8.5809221294602914796081766022, −8.1387067309874521210860884589, −7.33146545675343527744745605160, −6.463700464658224881340482404264, −5.65712391525408544494334620595, −4.98599005444184266963599912304, −3.91500733933555680119837021417, −3.18908638855021117982583282268, −2.82311563820905554767717498053, −1.96255002151170610481996546459, −0.57412484028412330542684005577, 0.285673079500002918882646241164, 1.1453222136772729154874948038, 2.20049947836678240561543756669, 2.6561216401746809333182443395, 3.466713504246229017175345439288, 4.9626036497677833865136725900, 5.43773825312656332602074302296, 6.40212340287061574463764158970, 6.91519243574882777125962986437, 7.41062442965093291542519599388, 8.00049759396778856151348174537, 8.859500724009768331425795110583, 9.40726968137655992338191786179, 10.18484152480240934012919756780, 10.69317094875324178213983201149, 11.64066370678739489876444471187, 12.303074263167120324361859506570, 12.89299842007639818426010197673, 13.568834490273698155117830723780, 14.315868349245149881179136600776, 15.16937503355102429395725871765, 15.43385257374648509016437024023, 16.63082796886813195787007872231, 16.89352135926619386846099052248, 17.45143195958945591790782263135

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