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Properties

Label 1-6025-6025.1423-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.364 - 0.931i$
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  + (−0.453 − 0.891i)2-s + (0.891 + 0.453i)3-s + (−0.587 + 0.809i)4-s − i·6-s + (−0.996 + 0.0784i)7-s + (0.987 + 0.156i)8-s + (0.587 + 0.809i)9-s + (0.0784 + 0.996i)11-s + (−0.891 + 0.453i)12-s + (0.760 + 0.649i)13-s + (0.522 + 0.852i)14-s + (−0.309 − 0.951i)16-s + (0.996 + 0.0784i)17-s + (0.453 − 0.891i)18-s + (−0.852 − 0.522i)19-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (0.891 + 0.453i)3-s + (−0.587 + 0.809i)4-s − i·6-s + (−0.996 + 0.0784i)7-s + (0.987 + 0.156i)8-s + (0.587 + 0.809i)9-s + (0.0784 + 0.996i)11-s + (−0.891 + 0.453i)12-s + (0.760 + 0.649i)13-s + (0.522 + 0.852i)14-s + (−0.309 − 0.951i)16-s + (0.996 + 0.0784i)17-s + (0.453 − 0.891i)18-s + (−0.852 − 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.364 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261707669 - 0.8610073616i\)
\(L(\frac12)\) \(\approx\) \(1.261707669 - 0.8610073616i\)
\(L(1)\) \(\approx\) \(1.006276674 - 0.2062319650i\)
\(L(1)\) \(\approx\) \(1.006276674 - 0.2062319650i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.453 - 0.891i)T \)
3 \( 1 + (0.891 + 0.453i)T \)
7 \( 1 + (-0.996 + 0.0784i)T \)
11 \( 1 + (0.0784 + 0.996i)T \)
13 \( 1 + (0.760 + 0.649i)T \)
17 \( 1 + (0.996 + 0.0784i)T \)
19 \( 1 + (-0.852 - 0.522i)T \)
23 \( 1 + (0.852 + 0.522i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (0.0784 - 0.996i)T \)
37 \( 1 + (0.233 - 0.972i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 + (0.0784 - 0.996i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (-0.453 - 0.891i)T \)
59 \( 1 + (-0.156 - 0.987i)T \)
61 \( 1 + (0.156 - 0.987i)T \)
67 \( 1 + (-0.891 - 0.453i)T \)
71 \( 1 + (0.649 - 0.760i)T \)
73 \( 1 + (0.760 - 0.649i)T \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.649 - 0.760i)T \)
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

−18.06019276158540496311954586308, −16.91973384661895299628351234399, −16.546561612418785944834027887729, −15.92615470383076860922141516831, −15.18890538832625005300903975960, −14.62328740334675663203586862792, −14.03766065106810991235956876933, −13.204461479997625490895528121527, −13.026849611074115528096982207308, −12.13947352780027118661709286011, −10.94208027943825519536926709695, −10.30687246254668250490525130539, −9.70339910453971772128452783042, −8.88822591724190897128057786669, −8.46422298526895033264381554527, −7.92279333085951028404090167735, −7.05295485906523904680279584195, −6.46083645647685262942487872077, −5.97473688752136000230582476010, −5.11817211042102185064326440501, −4.00260894383943719136257245026, −3.311739081724046680985050152368, −2.785808812735537816293908719638, −1.32374971537226674455833978829, −0.983742653484629570917072538065, 0.45878650226672648098611634585, 1.85956977316795720755712075595, 2.08083467630929817735330878564, 3.20871090695885043530677172840, 3.57572596767725801264647111239, 4.30047576193538675216253312847, 5.00065317600723364935842485365, 6.15835049156732678794304682721, 7.10677008721882463729753476871, 7.65075298414368874641666204620, 8.48662617542933545770428583423, 9.18669617906966044856501610215, 9.55299574118433808100911408126, 10.10688194144898047308687818759, 10.84182011340106564631220467173, 11.53021044952755505581218153756, 12.4478144752096126728100969090, 12.95550405398121710166274716766, 13.48090469296318290418930113465, 14.17196422467000582350261823050, 15.03993361219733363554061654061, 15.584306194773523205933624961296, 16.38491781067710518103663530308, 16.935679141863490259327758774296, 17.59684720564240193771809647005

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