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Properties

Label 1-6025-6025.622-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.121 + 0.992i$
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  + (−0.156 − 0.987i)2-s + (−0.987 − 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (−0.852 + 0.522i)7-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.522 − 0.852i)11-s + (0.987 − 0.156i)12-s + (0.233 − 0.972i)13-s + (0.649 + 0.760i)14-s + (0.809 − 0.587i)16-s + (0.852 + 0.522i)17-s + (0.156 − 0.987i)18-s + (0.760 + 0.649i)19-s + ⋯
L(s)  = 1  + (−0.156 − 0.987i)2-s + (−0.987 − 0.156i)3-s + (−0.951 + 0.309i)4-s + i·6-s + (−0.852 + 0.522i)7-s + (0.453 + 0.891i)8-s + (0.951 + 0.309i)9-s + (−0.522 − 0.852i)11-s + (0.987 − 0.156i)12-s + (0.233 − 0.972i)13-s + (0.649 + 0.760i)14-s + (0.809 − 0.587i)16-s + (0.852 + 0.522i)17-s + (0.156 − 0.987i)18-s + (0.760 + 0.649i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.121 + 0.992i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.121 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0009257074593 + 0.001045613037i\)
\(L(\frac12)\) \(\approx\) \(0.0009257074593 + 0.001045613037i\)
\(L(1)\) \(\approx\) \(0.4687017682 - 0.2573480648i\)
\(L(1)\) \(\approx\) \(0.4687017682 - 0.2573480648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.156 - 0.987i)T \)
3 \( 1 + (-0.987 - 0.156i)T \)
7 \( 1 + (-0.852 + 0.522i)T \)
11 \( 1 + (-0.522 - 0.852i)T \)
13 \( 1 + (0.233 - 0.972i)T \)
17 \( 1 + (0.852 + 0.522i)T \)
19 \( 1 + (0.760 + 0.649i)T \)
23 \( 1 + (-0.760 - 0.649i)T \)
29 \( 1 + (-0.156 - 0.987i)T \)
31 \( 1 + (-0.522 + 0.852i)T \)
37 \( 1 + (-0.996 - 0.0784i)T \)
41 \( 1 + (-0.707 + 0.707i)T \)
43 \( 1 + (-0.522 + 0.852i)T \)
47 \( 1 + (0.987 - 0.156i)T \)
53 \( 1 + (-0.156 - 0.987i)T \)
59 \( 1 + (0.891 + 0.453i)T \)
61 \( 1 + (-0.891 + 0.453i)T \)
67 \( 1 + (0.987 + 0.156i)T \)
71 \( 1 + (0.972 + 0.233i)T \)
73 \( 1 + (0.233 + 0.972i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.972 + 0.233i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−18.18417068982463071723589958312, −17.180770254807965481101572860095, −16.97507576242803897591868456176, −16.18449492618513801261506127329, −15.74612620023348686689785301626, −15.30453990067798264082278221296, −14.17981343480481344363652418895, −13.72932254519195851623089434570, −12.99750944448512084197759342442, −12.31112629305297527219955440117, −11.71466293928216065422864703722, −10.64790171010367192304830298907, −10.180300065665707824401093862075, −9.44307658022367109311840879794, −9.141191264736951635115036181242, −7.80162738450117354684505931763, −7.185433108664887486069789472127, −6.8987470641108746037137341381, −6.06610819496551706159159224887, −5.32318698596547080627827125417, −4.88246581735075571718146340793, −3.93924928988503507100719689478, −3.457050299200510285022289107642, −1.93505576542571081369157709729, −0.98395807053571322977368375072, 0.00064905786288854641681025503, 0.85210040267956921920531358265, 1.686478753979375459657109881609, 2.69743684322578208340829992305, 3.38911099128340605759660588894, 3.961931152311539740170486115009, 5.1604425443215052877400867559, 5.595416942544678620360709176601, 6.09609986193287774275009083070, 7.14259048606606003166461509771, 8.14559779332302953785936774284, 8.411139832221153638086652948218, 9.64576515647078441685568102388, 10.09188272136406150796772205585, 10.54374428703458196836277863798, 11.33156620421130613298889891589, 11.99791330121016341568375494295, 12.49583652157494733830451063791, 13.017150740374376994809990451175, 13.62611272801258411663429396852, 14.44570439083187392736572981055, 15.488174820232876598234887080499, 16.11239359925210074172415724041, 16.653088223598079351725635212723, 17.31659283001704777220234936370

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