Properties

Label 1-6025-6025.3172-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.417 - 0.908i$
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  + (0.629 + 0.777i)2-s + (−0.777 − 0.629i)3-s + (−0.207 + 0.978i)4-s i·6-s + (0.284 + 0.958i)7-s + (−0.891 + 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.958 + 0.284i)11-s + (0.777 − 0.629i)12-s + (0.999 − 0.0261i)13-s + (−0.566 + 0.824i)14-s + (−0.913 − 0.406i)16-s + (0.972 − 0.233i)17-s + (−0.629 + 0.777i)18-s + (−0.824 + 0.566i)19-s + ⋯
L(s)  = 1  + (0.629 + 0.777i)2-s + (−0.777 − 0.629i)3-s + (−0.207 + 0.978i)4-s i·6-s + (0.284 + 0.958i)7-s + (−0.891 + 0.453i)8-s + (0.207 + 0.978i)9-s + (−0.958 + 0.284i)11-s + (0.777 − 0.629i)12-s + (0.999 − 0.0261i)13-s + (−0.566 + 0.824i)14-s + (−0.913 − 0.406i)16-s + (0.972 − 0.233i)17-s + (−0.629 + 0.777i)18-s + (−0.824 + 0.566i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (3172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.417 - 0.908i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3754157065 - 0.2407609573i\)
\(L(\frac12)\) \(\approx\) \(0.3754157065 - 0.2407609573i\)
\(L(1)\) \(\approx\) \(0.8614789543 + 0.3794984737i\)
\(L(1)\) \(\approx\) \(0.8614789543 + 0.3794984737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.629 + 0.777i)T \)
3 \( 1 + (-0.777 - 0.629i)T \)
7 \( 1 + (0.284 + 0.958i)T \)
11 \( 1 + (-0.958 + 0.284i)T \)
13 \( 1 + (0.999 - 0.0261i)T \)
17 \( 1 + (0.972 - 0.233i)T \)
19 \( 1 + (-0.824 + 0.566i)T \)
23 \( 1 + (-0.0784 - 0.996i)T \)
29 \( 1 + (-0.629 - 0.777i)T \)
31 \( 1 + (0.725 - 0.688i)T \)
37 \( 1 + (-0.333 + 0.942i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (-0.233 - 0.972i)T \)
47 \( 1 + (0.156 + 0.987i)T \)
53 \( 1 + (0.358 - 0.933i)T \)
59 \( 1 + (-0.998 + 0.0523i)T \)
61 \( 1 + (-0.453 - 0.891i)T \)
67 \( 1 + (-0.933 + 0.358i)T \)
71 \( 1 + (-0.878 - 0.477i)T \)
73 \( 1 + (-0.522 + 0.852i)T \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.878 - 0.477i)T \)
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

−17.92535618720215235768751644387, −17.21423572507799567627530114534, −16.371793513573607803447183609882, −15.89261729318890868438418037703, −15.14575384669662878995209469029, −14.558990648165347519730005783789, −13.65443533312015149567288291660, −13.30381631552040836767059573438, −12.46829714630687382036665615754, −11.86328332676780863217685767942, −10.94358898937360816937245529205, −10.7472404951638652854635281137, −10.3044831665118750749509491903, −9.4065984138786380680071082906, −8.72969197559217855289564446770, −7.69244462740076571820906499940, −6.86370621028265219756460680268, −5.99411980912485562104002446196, −5.5024925604227263924488355338, −4.81694692687872375155165567843, −4.07833121636184945051421820169, −3.557960245741379599495102248104, −2.845550155683233899229080065720, −1.55344307895310641373624533957, −0.99192379650455536609687706098, 0.10899988034190450647142764690, 1.49678175136544269770352579453, 2.39039215378729786614759465816, 3.025079315262762929303592035227, 4.214455181896313812170088092120, 4.79620274707329556115942294701, 5.63734339485935241235900672454, 5.95263814878795608266976576735, 6.56201804526820493430960327794, 7.49728729255862474092071426549, 8.164308045132290339702742945981, 8.39897514839538417404590181998, 9.55652640218752218831986719606, 10.46442525376929939509769033590, 11.21284463665460588948982658416, 11.91915048210283651111141905567, 12.40149081639019834817174097655, 12.998591289790621599643973461075, 13.54544780767614740208997838133, 14.32642310181600148235423632863, 15.09200238658315275681855794763, 15.61048573534438374661143467789, 16.30454154854834801556209986386, 16.81797191427317487941430567236, 17.54892334582309986938634718931

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