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Properties

Label 1-6025-6025.377-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.539 + 0.842i$
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 + (0.156 − 0.987i)3-s + (−0.587 + 0.809i)4-s + (0.951 − 0.309i)6-s + (0.852 + 0.522i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (−0.0784 − 0.996i)11-s + (0.707 + 0.707i)12-s + (−0.382 + 0.923i)13-s + (−0.0784 + 0.996i)14-s + (−0.309 − 0.951i)16-s + (−0.760 − 0.649i)17-s + (−0.156 − 0.987i)18-s + (0.852 + 0.522i)19-s + ⋯
L(s)  = 1  + (0.453 + 0.891i)2-s + (0.156 − 0.987i)3-s + (−0.587 + 0.809i)4-s + (0.951 − 0.309i)6-s + (0.852 + 0.522i)7-s + (−0.987 − 0.156i)8-s + (−0.951 − 0.309i)9-s + (−0.0784 − 0.996i)11-s + (0.707 + 0.707i)12-s + (−0.382 + 0.923i)13-s + (−0.0784 + 0.996i)14-s + (−0.309 − 0.951i)16-s + (−0.760 − 0.649i)17-s + (−0.156 − 0.987i)18-s + (0.852 + 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.539 + 0.842i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7656129074 + 1.399537962i\)
\(L(\frac12)\) \(\approx\) \(0.7656129074 + 1.399537962i\)
\(L(1)\) \(\approx\) \(1.148750008 + 0.4149806135i\)
\(L(1)\) \(\approx\) \(1.148750008 + 0.4149806135i\)

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 \( 1 + (0.156 - 0.987i)T \)
7 \( 1 + (0.852 + 0.522i)T \)
11 \( 1 + (-0.0784 - 0.996i)T \)
13 \( 1 + (-0.382 + 0.923i)T \)
17 \( 1 + (-0.760 - 0.649i)T \)
19 \( 1 + (0.852 + 0.522i)T \)
23 \( 1 + (-0.233 + 0.972i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (0.649 - 0.760i)T \)
37 \( 1 + (-0.852 - 0.522i)T \)
41 \( 1 + (0.156 + 0.987i)T \)
43 \( 1 + (0.522 + 0.852i)T \)
47 \( 1 + (-0.453 + 0.891i)T \)
53 \( 1 + (-0.891 - 0.453i)T \)
59 \( 1 + (-0.987 - 0.156i)T \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (-0.453 - 0.891i)T \)
71 \( 1 + (0.0784 - 0.996i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (-0.891 + 0.453i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
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

−17.516497562465251500059619619713, −17.143324031874100433250267411216, −15.70890232394496971488665738819, −15.55039639737832696905963729282, −14.74034807725446445524039655330, −14.22834183447253263475220715966, −13.66257630093072109545426686199, −12.83176721731200139296069096308, −12.087431179772965188722911372182, −11.5297620851895689287046851188, −10.69085311527081782983583224743, −10.260044948485003875381870575248, −9.94373856909755152333222172178, −8.81351456284789085594138078140, −8.4801981866065722518124761585, −7.49359590473419157010304605838, −6.56649703217107084010698119942, −5.51311863230795128975207297388, −4.9535383400001308064586926115, −4.45627237355978317734400653237, −3.88846111446522212306541700683, −2.92530833899956767296434579398, −2.373214275817035335977163613919, −1.47131675035344977558369022276, −0.35569601359947434706827877381, 0.9964242281334989719595361889, 1.9264516989360187899521504899, 2.80709615673499638233023614257, 3.41667103242859991591020068619, 4.55252709245719589579805955496, 5.09818929862655629025841923757, 5.94953612493870099123531861373, 6.36661816024648824150775608349, 7.21532630632824460495876356658, 7.86196999603776994207911742680, 8.25724141031951790460404939661, 9.11121661915159929860730313114, 9.495183701913352660113153396156, 11.05698601641211413941293799392, 11.622876613962443960691964524954, 12.00753624841500580079869216184, 12.819146789688710907272686552121, 13.5880492591600778570392063811, 14.15180132210660939340281621745, 14.294437106807908424545165026303, 15.30299479012408609267510462726, 15.946791136494283486048210453427, 16.5771102455481135666585423221, 17.35407491757002591142891690255, 17.96953886077569143334611577489

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