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Properties

Label 1-6025-6025.4253-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.823 - 0.566i$
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  + (−0.987 + 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s − i·6-s + (−0.233 + 0.972i)7-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (0.972 + 0.233i)11-s + (0.156 + 0.987i)12-s + (0.852 − 0.522i)13-s + (0.0784 − 0.996i)14-s + (0.809 − 0.587i)16-s + (0.233 + 0.972i)17-s + (0.987 + 0.156i)18-s + (0.996 − 0.0784i)19-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (−0.156 + 0.987i)3-s + (0.951 − 0.309i)4-s − i·6-s + (−0.233 + 0.972i)7-s + (−0.891 + 0.453i)8-s + (−0.951 − 0.309i)9-s + (0.972 + 0.233i)11-s + (0.156 + 0.987i)12-s + (0.852 − 0.522i)13-s + (0.0784 − 0.996i)14-s + (0.809 − 0.587i)16-s + (0.233 + 0.972i)17-s + (0.987 + 0.156i)18-s + (0.996 − 0.0784i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.823 - 0.566i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (4253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.823 - 0.566i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1605683583 + 0.5169056128i\)
\(L(\frac12)\) \(\approx\) \(-0.1605683583 + 0.5169056128i\)
\(L(1)\) \(\approx\) \(0.5314688812 + 0.3459637660i\)
\(L(1)\) \(\approx\) \(0.5314688812 + 0.3459637660i\)

Euler product

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

−17.36536333385774859886755582773, −16.85645222000649393920440517302, −16.20875147500500959971469206633, −15.77909833511615929424117021116, −14.45006245917221345704587435867, −13.97790387583690550266030700091, −13.454562806963056206588477193287, −12.51523315014436527597891211744, −11.869291909764673793870711241192, −11.35680843224393909324956091786, −10.86980283616259145249722085036, −9.80565982992417487171503413534, −9.42560410576270965038573041213, −8.500399772642172716591750048790, −7.95437163541731355325464576474, −7.19447826941364452207582240888, −6.753958062981091483622500495193, −6.17467263767521261483225097022, −5.33052217703397459852376593366, −3.984334527996872067464188251364, −3.42078704382513158641069681632, −2.53995929285571018291080211534, −1.50128586827446793554555355596, −1.160638756271987741490201220421, −0.21399425142116533614441608031, 1.15952244719395454076030469625, 1.93653723523242384383729402151, 3.00699167869496579643185565952, 3.50215544051125158942150576634, 4.42017222411730510446542711596, 5.5969661365026207593222048920, 5.86040613493970444127320650617, 6.4997595997506606157877908429, 7.547197201833524435037230091079, 8.404706210769205473424437072055, 8.77848237591053243321095156754, 9.52853917660265198746571265175, 9.93486404416770161536684741468, 10.71112879357516441292827363441, 11.34038848496725310867975806862, 12.00884632170924305964387256863, 12.4353724303538557333179349084, 13.747892946498144155816801123967, 14.44763953376117760577384337117, 15.23865983093497813404810467378, 15.51526366259958096280031775062, 16.1304577570786684407363315580, 16.78736700850227137812007116896, 17.47017977019435012960247623578, 17.8978310164451760629965423435

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