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Properties

Label 1-6025-6025.62-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.797 + 0.603i$
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  + (−0.838 + 0.544i)2-s + (−0.0523 − 0.998i)3-s + (0.406 − 0.913i)4-s + (0.587 + 0.809i)6-s + (−0.477 − 0.878i)7-s + (0.156 + 0.987i)8-s + (−0.994 + 0.104i)9-s + (−0.182 − 0.983i)11-s + (−0.933 − 0.358i)12-s + (−0.477 + 0.878i)13-s + (0.878 + 0.477i)14-s + (−0.669 − 0.743i)16-s + (−0.923 − 0.382i)17-s + (0.777 − 0.629i)18-s + (0.688 − 0.725i)19-s + ⋯
L(s)  = 1  + (−0.838 + 0.544i)2-s + (−0.0523 − 0.998i)3-s + (0.406 − 0.913i)4-s + (0.587 + 0.809i)6-s + (−0.477 − 0.878i)7-s + (0.156 + 0.987i)8-s + (−0.994 + 0.104i)9-s + (−0.182 − 0.983i)11-s + (−0.933 − 0.358i)12-s + (−0.477 + 0.878i)13-s + (0.878 + 0.477i)14-s + (−0.669 − 0.743i)16-s + (−0.923 − 0.382i)17-s + (0.777 − 0.629i)18-s + (0.688 − 0.725i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.797 + 0.603i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (62, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.797 + 0.603i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1539661542 - 0.4581252907i\)
\(L(\frac12)\) \(\approx\) \(-0.1539661542 - 0.4581252907i\)
\(L(1)\) \(\approx\) \(0.5190418988 - 0.2528174361i\)
\(L(1)\) \(\approx\) \(0.5190418988 - 0.2528174361i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.838 + 0.544i)T \)
3 \( 1 + (-0.0523 - 0.998i)T \)
7 \( 1 + (-0.477 - 0.878i)T \)
11 \( 1 + (-0.182 - 0.983i)T \)
13 \( 1 + (-0.477 + 0.878i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.688 - 0.725i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (0.608 + 0.793i)T \)
37 \( 1 + (0.130 - 0.991i)T \)
41 \( 1 + (-0.453 + 0.891i)T \)
43 \( 1 + (0.852 - 0.522i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.965 - 0.258i)T \)
59 \( 1 + (0.965 - 0.258i)T \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + (-0.258 - 0.965i)T \)
71 \( 1 + (-0.608 - 0.793i)T \)
73 \( 1 + (-0.522 + 0.852i)T \)
79 \( 1 + (-0.156 - 0.987i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (0.566 - 0.824i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−17.94151063114195159984929724513, −17.53550739691838637541887243416, −16.91355264995903711118625503350, −16.1056805400614606326769468266, −15.56462867217754879965870849292, −15.15194479590293000087362405726, −14.4451211383107860952927178828, −13.1320166640560863082042786759, −12.78083427918067603264940749565, −11.94768968951409625134381256206, −11.45708524205449838790185124396, −10.62904450351991159939698328954, −10.035253890250048134965681781802, −9.59568425531963501899203797764, −8.9689531706394667069020323087, −8.335964632646015677932314826430, −7.573643001795719584457279653194, −6.756182112883808683338430062328, −5.865470738966765836957272943174, −5.12157794200287589460430033239, −4.36059837017765068429861398384, −3.47767283072555635537421286703, −2.82763975163458045262167982827, −2.29568589331918621240321041571, −1.19282946438535987583402736075, 0.2290483895496660895298716367, 0.74844280106338704509084141868, 1.71094947581928732146175091351, 2.55324989170996309871432878202, 3.255744461228289700744902517650, 4.55192171372789250039483126367, 5.23388334500114486075879333710, 6.23324575397739419067767580256, 6.64179890960288973497564574196, 7.220092164930808901456363593477, 7.754359463669748546318865387314, 8.65058819223605056173923651843, 9.10210527419542341670382483230, 9.86344730259179012345558521354, 10.763984940764099046630105051867, 11.324173285122733821516461220563, 11.76697049320353609726618635829, 12.955831526528981467804672725994, 13.45933308413589870569395468588, 14.04888699143590556188547061758, 14.5529278444325706419789353968, 15.59963905806638949292355027431, 16.232083479056454162092007171689, 16.711702710681660024404727000533, 17.48222978266049376492020442668

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