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Properties

Label 1-6025-6025.2612-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.982 - 0.188i$
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  + (−0.0523 − 0.998i)2-s + (0.358 − 0.933i)3-s + (−0.994 + 0.104i)4-s + (−0.951 − 0.309i)6-s + (−0.284 − 0.958i)7-s + (0.156 + 0.987i)8-s + (−0.743 − 0.669i)9-s + (−0.942 − 0.333i)11-s + (−0.258 + 0.965i)12-s + (0.793 − 0.608i)13-s + (−0.942 + 0.333i)14-s + (0.978 − 0.207i)16-s + (−0.0784 + 0.996i)17-s + (−0.629 + 0.777i)18-s + (−0.284 − 0.958i)19-s + ⋯
L(s)  = 1  + (−0.0523 − 0.998i)2-s + (0.358 − 0.933i)3-s + (−0.994 + 0.104i)4-s + (−0.951 − 0.309i)6-s + (−0.284 − 0.958i)7-s + (0.156 + 0.987i)8-s + (−0.743 − 0.669i)9-s + (−0.942 − 0.333i)11-s + (−0.258 + 0.965i)12-s + (0.793 − 0.608i)13-s + (−0.942 + 0.333i)14-s + (0.978 − 0.207i)16-s + (−0.0784 + 0.996i)17-s + (−0.629 + 0.777i)18-s + (−0.284 − 0.958i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2612, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.982 - 0.188i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2697825575 + 0.02570601170i\)
\(L(\frac12)\) \(\approx\) \(0.2697825575 + 0.02570601170i\)
\(L(1)\) \(\approx\) \(0.4999095755 - 0.6537653531i\)
\(L(1)\) \(\approx\) \(0.4999095755 - 0.6537653531i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.0523 - 0.998i)T \)
3 \( 1 + (0.358 - 0.933i)T \)
7 \( 1 + (-0.284 - 0.958i)T \)
11 \( 1 + (-0.942 - 0.333i)T \)
13 \( 1 + (0.793 - 0.608i)T \)
17 \( 1 + (-0.0784 + 0.996i)T \)
19 \( 1 + (-0.284 - 0.958i)T \)
23 \( 1 + (0.522 - 0.852i)T \)
29 \( 1 + (0.544 - 0.838i)T \)
31 \( 1 + (0.430 + 0.902i)T \)
37 \( 1 + (0.284 + 0.958i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (0.233 + 0.972i)T \)
47 \( 1 + (-0.891 + 0.453i)T \)
53 \( 1 + (0.998 + 0.0523i)T \)
59 \( 1 + (0.777 - 0.629i)T \)
61 \( 1 + (-0.156 + 0.987i)T \)
67 \( 1 + (0.0523 + 0.998i)T \)
71 \( 1 + (-0.182 + 0.983i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (-0.453 + 0.891i)T \)
83 \( 1 + (-0.913 - 0.406i)T \)
89 \( 1 + (-0.430 - 0.902i)T \)
97 \( 1 + (-0.669 - 0.743i)T \)
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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

−17.71938244606647098779103323380, −16.74102175011649103377481653896, −16.202093526400706509994483379488, −15.86772759948844876033788393586, −15.08570168464033647291321711423, −14.86469591391188605502043197642, −13.78526049458853961207467883144, −13.53634071619498284317631477438, −12.60871478823187126082664084410, −11.81524280294951957932376333191, −10.97188336918670225135159560268, −10.132450009053734745917723310438, −9.61283713452469381864777228775, −8.95601893072363603825304707294, −8.47941152092429131068870126120, −7.7823004628294724990515264116, −6.97490011337522937361036437305, −6.10484826155846163664356572175, −5.41023973160142582488274524467, −5.03093649082273954088831796840, −4.13486020259233113265244541658, −3.43140560155864821113158879930, −2.66207663591617921906642725204, −1.67893737407063404363518048143, −0.07593484513449711075262763382, 0.948734800141855739841060738783, 1.36640016210982403104155967856, 2.67287258486817074564423167148, 2.84653183351515305811570403896, 3.77048212151559638969057395816, 4.49950265707856573469536936987, 5.42017642335731877250339726363, 6.29404525478910035140104875076, 6.93196569833063893433073088624, 7.91983433652509394147835553684, 8.38359420735289282504976380733, 8.81998240623317504188914122853, 10.05741581445080902118056919165, 10.35193851055109017492796324650, 11.15005832694011462878718312952, 11.66045597423243403973250350221, 12.77677546227012177381743014953, 13.02504240211202160320834821583, 13.41481467202246939911725203587, 14.11469486776710980785836358273, 14.84720450501987895735825566791, 15.661254423752131704989450970910, 16.61467065717577828935809529986, 17.44976766945843998297818474533, 17.698670897826918367153187747907

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