Properties

Label 1-6025-6025.2856-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.237 + 0.971i$
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  + (−0.809 + 0.587i)2-s + 3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.309 + 0.951i)8-s + 9-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 + 0.587i)18-s + (0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + 3-s + (0.309 − 0.951i)4-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + (0.309 + 0.951i)8-s + 9-s + (−0.809 + 0.587i)11-s + (0.309 − 0.951i)12-s + (0.309 − 0.951i)13-s + 14-s + (−0.809 − 0.587i)16-s + (0.309 − 0.951i)17-s + (−0.809 + 0.587i)18-s + (0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.237 + 0.971i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2856, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.237 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9700048689 + 0.7616413853i\)
\(L(\frac12)\) \(\approx\) \(0.9700048689 + 0.7616413853i\)
\(L(1)\) \(\approx\) \(0.8791436566 + 0.1850376100i\)
\(L(1)\) \(\approx\) \(0.8791436566 + 0.1850376100i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + T \)
7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
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 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
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 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.809 + 0.587i)T \)
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

−17.850960527037605585989680034321, −16.91137137135788361615182216610, −16.1755310164650137603516501383, −15.649009985128767089304928810960, −15.302556089191302829144643620673, −14.02379057948386460370992966037, −13.584423905718747871566424009098, −12.97356177521318685686028014912, −12.23907019087040234850907979369, −11.70315951437576579497083451367, −10.69587684304067505002133964968, −10.1735650721855289093724907293, −9.532709844764878406359081510522, −8.81211553913464649711882530673, −8.52442346959834184655339319902, −7.73198220956734191863475943521, −6.978499443785462029010331278786, −6.34398646273911699771515531273, −5.334090999577177357405709572598, −4.16926262065269081007513119563, −3.53639769470686986934960740601, −2.98868528324952971091651473550, −2.18173932564723109770619215106, −1.681145646063402959085163848626, −0.415819460602233752193692679846, 0.84079557334110703892995582108, 1.61888199706590220039616357331, 2.715161824510204789018723331638, 3.08547678535178409086947605654, 4.18143096007420493484543256846, 4.9618269193203215236536610256, 5.845028484699436164929060977882, 6.62188033521477141324668300517, 7.302022370226196942077862691996, 7.97589165355934668944417673038, 8.18035676305079255623450535825, 9.37217743583943620712366952789, 9.676751280243782911919533174947, 10.369823700057203815798837646741, 10.72020940412310292798062394741, 12.04811724312012486175913128931, 12.70778040632468894136324446215, 13.50138426769890497539058967600, 13.97739133493573908850239098381, 14.69504639393228763560648044565, 15.37345608482603565758957976783, 15.95542794924293284596110709832, 16.32573947974851809322125309223, 17.15427436228208996634892095509, 18.077075647359386063523695530614

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