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Properties

Label 1-6025-6025.4937-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.998 + 0.0507i$
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 + (0.453 − 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (0.972 − 0.233i)7-s + (−0.987 + 0.156i)8-s + (−0.587 − 0.809i)9-s + (0.996 + 0.0784i)11-s + (−0.987 + 0.156i)12-s + (0.972 + 0.233i)13-s + (0.233 − 0.972i)14-s + (−0.309 + 0.951i)16-s + (0.923 − 0.382i)17-s + (−0.987 + 0.156i)18-s + (−0.522 − 0.852i)19-s + ⋯
L(s)  = 1  + (0.453 − 0.891i)2-s + (0.453 − 0.891i)3-s + (−0.587 − 0.809i)4-s + (−0.587 − 0.809i)6-s + (0.972 − 0.233i)7-s + (−0.987 + 0.156i)8-s + (−0.587 − 0.809i)9-s + (0.996 + 0.0784i)11-s + (−0.987 + 0.156i)12-s + (0.972 + 0.233i)13-s + (0.233 − 0.972i)14-s + (−0.309 + 0.951i)16-s + (0.923 − 0.382i)17-s + (−0.987 + 0.156i)18-s + (−0.522 − 0.852i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.998 + 0.0507i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.998 + 0.0507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08341500644 - 3.283277329i\)
\(L(\frac12)\) \(\approx\) \(-0.08341500644 - 3.283277329i\)
\(L(1)\) \(\approx\) \(1.011771317 - 1.424292235i\)
\(L(1)\) \(\approx\) \(1.011771317 - 1.424292235i\)

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 \( 1 + (0.453 - 0.891i)T \)
7 \( 1 + (0.972 - 0.233i)T \)
11 \( 1 + (0.996 + 0.0784i)T \)
13 \( 1 + (0.972 + 0.233i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
19 \( 1 + (-0.522 - 0.852i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (0.923 + 0.382i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (0.233 - 0.972i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
61 \( 1 + (0.707 + 0.707i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (-0.382 - 0.923i)T \)
73 \( 1 + (0.972 - 0.233i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.760 - 0.649i)T \)
97 \( 1 + T \)
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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

−17.84140843400735437572557095650, −17.06836638233732253303366892021, −16.714526123625686811515584503113, −15.93876368653353804861544116663, −15.410111588128586045320523083418, −14.65549598389121450871711011203, −14.30155796869804650863625219091, −13.92103663928365092379038667270, −12.8786060754126277219829181528, −12.25097293473658131655819661412, −11.33969480771245421886712956118, −10.92505922462943632952249159890, −9.81373344833740102484870770598, −9.27524669191645944679137707952, −8.55907608595269706695753068187, −7.93644358378204086239300112403, −7.63990135285452864121182087951, −6.22358373961155781098514836614, −5.88553439928354922000457211892, −5.19379468064717830237236938643, −4.11550124525106511255275619561, −4.04660716730499012390489072330, −3.21277940943031168912807588465, −2.17157062569935959281497658086, −1.1898828743626391224829252680, 0.72510624642328277555978449907, 1.279475144953271639881630355927, 1.99635423719522677634041233243, 2.65904252902804730855810125273, 3.66978823302731437097221063595, 4.09831256802156059758471627994, 4.97474744640854940292531470304, 5.92771484410844037640907111666, 6.41057443645081812073465078761, 7.31657830957331278220652183326, 8.12197343431394839284355745188, 8.775961268322692861212022154017, 9.28191668039811656411094768010, 10.202432986794896311042285203957, 10.99518535964141489285607233377, 11.709238377000575373270371677781, 11.911115145459128420037881071968, 12.79747437487133689861488817264, 13.50745184165087293246881025926, 13.9591896754712324988716712326, 14.5427862377776382042372402160, 14.96026640340904221948094567740, 15.95124500624615941688511115835, 17.04014798087820773718992710608, 17.581146009279323861280206036461

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