Properties

Degree $1$
Conductor $6025$
Sign $0.323 + 0.946i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.891 − 0.453i)2-s + (−0.891 − 0.453i)3-s + (0.587 + 0.809i)4-s + (0.587 + 0.809i)6-s + (0.522 − 0.852i)7-s + (−0.156 − 0.987i)8-s + (0.587 + 0.809i)9-s + (−0.760 + 0.649i)11-s + (−0.156 − 0.987i)12-s + (0.522 + 0.852i)13-s + (−0.852 + 0.522i)14-s + (−0.309 + 0.951i)16-s + (0.923 + 0.382i)17-s + (−0.156 − 0.987i)18-s + (−0.972 − 0.233i)19-s + ⋯
L(s,χ)  = 1  + (−0.891 − 0.453i)2-s + (−0.891 − 0.453i)3-s + (0.587 + 0.809i)4-s + (0.587 + 0.809i)6-s + (0.522 − 0.852i)7-s + (−0.156 − 0.987i)8-s + (0.587 + 0.809i)9-s + (−0.760 + 0.649i)11-s + (−0.156 − 0.987i)12-s + (0.522 + 0.852i)13-s + (−0.852 + 0.522i)14-s + (−0.309 + 0.951i)16-s + (0.923 + 0.382i)17-s + (−0.156 − 0.987i)18-s + (−0.972 − 0.233i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.323 + 0.946i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.323 + 0.946i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.323 + 0.946i$
Motivic weight: \(0\)
Character: $\chi_{6025} (213, \cdot )$
Sato-Tate group: $\mu(80)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.323 + 0.946i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.3498895283 + 0.2502546045i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.3498895283 + 0.2502546045i\)
\(L(\chi,1)\) \(\approx\) \(0.5100375896 - 0.1073770478i\)
\(L(1,\chi)\) \(\approx\) \(0.5100375896 - 0.1073770478i\)

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

−17.57100009484121045895081745863, −16.90373353200317651124159652259, −16.35575698687611902921769517552, −15.664906159500999666261289303697, −15.3089299261375992196364784125, −14.64271530654336511957200673579, −13.79886390355600353559832030153, −12.78751908374415694797320851633, −12.058808163105014093240750387985, −11.49029478147368599951098237654, −10.79878178636305429910858843743, −10.28144201447777696618221854152, −9.75425718457701892364580672710, −8.74885079930288498041075047234, −8.29345558372968480154713242585, −7.69263569571674526161805471409, −6.67789224341620604447927183634, −5.96716991842301793872839380956, −5.51289152268036658828033353110, −5.06145195575054170211220002519, −3.94713879346409766839187322661, −2.93344478266273266183945110184, −2.08024570370364173378743320661, −1.10658875201673332974168563811, −0.22129675200937336083311785801, 0.852939964961738612302979461591, 1.72844148396214840307109026978, 2.05053903976259797433170646061, 3.31992176983222117337372474620, 4.24167306509232684437457727405, 4.71659051390743954222221858766, 5.901327703378367924449776069990, 6.472305511202659085394379751037, 7.2715057196014392615004952171, 7.82739807353472664038074494561, 8.22527986214481416607066078929, 9.40685999700969221840089893513, 10.027153245656166631250050734957, 10.61382664390427101809839509575, 11.21399405024252236174081263080, 11.636875797933162614455758510864, 12.54344358094553579134557005662, 12.95977308016989215073351716117, 13.67269802302174391876063860634, 14.58451886687443471541609846265, 15.48130611497697643046318642732, 16.231132184175239510523245778874, 16.779782254105633573009865881606, 17.22545460817875733798147331859, 17.83151040643438853031049709044

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