Properties

Degree $1$
Conductor $6025$
Sign $-0.823 + 0.566i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.823 + 0.566i$
Motivic weight: \(0\)
Character: $\chi_{6025} (17, \cdot )$
Sato-Tate group: $\mu(80)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.823 + 0.566i)\)

Particular Values

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

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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