Properties

Degree $1$
Conductor $6025$
Sign $-0.367 - 0.929i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.933 − 0.358i)2-s + (−0.838 + 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (0.333 − 0.942i)7-s + (0.453 − 0.891i)8-s + (0.406 − 0.913i)9-s + (−0.0261 + 0.999i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (−0.0261 − 0.999i)14-s + (0.104 − 0.994i)16-s + (0.972 − 0.233i)17-s + (0.0523 − 0.998i)18-s + (0.333 − 0.942i)19-s + ⋯
L(s,χ)  = 1  + (0.933 − 0.358i)2-s + (−0.838 + 0.544i)3-s + (0.743 − 0.669i)4-s + (−0.587 + 0.809i)6-s + (0.333 − 0.942i)7-s + (0.453 − 0.891i)8-s + (0.406 − 0.913i)9-s + (−0.0261 + 0.999i)11-s + (−0.258 + 0.965i)12-s + (−0.793 + 0.608i)13-s + (−0.0261 − 0.999i)14-s + (0.104 − 0.994i)16-s + (0.972 − 0.233i)17-s + (0.0523 − 0.998i)18-s + (0.333 − 0.942i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.367 - 0.929i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.367 - 0.929i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.367 - 0.929i$
Motivic weight: \(0\)
Character: $\chi_{6025} (42, \cdot )$
Sato-Tate group: $\mu(240)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.367 - 0.929i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.310026807 - 1.926973703i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.310026807 - 1.926973703i\)
\(L(\chi,1)\) \(\approx\) \(1.387720236 - 0.4878903063i\)
\(L(1,\chi)\) \(\approx\) \(1.387720236 - 0.4878903063i\)

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

−17.9299209018713583896509192415, −16.96418141211662048297417727081, −16.71503883653462169585504166104, −15.83257861997511473566590190559, −15.43054262568331173871807003903, −14.408003036471332037752269560323, −14.12779713762968246760997679951, −13.20729461521509343469227991006, −12.55590456928396853239377459064, −12.072414125370632945568539590865, −11.677422098013711461240241488634, −10.81773379316854402073852616159, −10.29284700111745027715268997063, −9.11825282459257969934947725205, −8.22777516675000918790554454239, −7.64824496782186832472699154567, −7.16551714014139402079276361212, −6.03295540462078258002915224576, −5.63604552796669210795158210392, −5.40057653211624922421771126100, −4.46339309342839595005358151064, −3.44215348894478726098842056719, −2.82011231225308851998241801338, −1.90028812498719889305387748725, −1.138310497467197391860897441904, 0.48249715377818707121239178912, 1.32436225406848720026658719496, 2.27781943536521057890755278916, 3.13925936109697898060986919806, 4.100544116873770947355606917484, 4.625369290621564432217723733924, 4.866440186868152143543335632314, 5.88668244273483777648055749199, 6.5535388457016050545115457185, 7.26728205868099798441974753850, 7.71732656148526330727427714338, 9.28659315933176835884370456917, 9.82356886075781839252689880663, 10.28389416972885227290463066320, 11.062360237772725609591625121067, 11.569009716497973499086875276036, 12.23746337470000278729065873088, 12.71344288399507850924373808634, 13.609480624750443320765033041832, 14.25548177464459392877561780990, 14.877936701127482099205297284435, 15.38659059545560491843535492914, 16.22454615225095341674085028709, 16.77206920766676533497490299824, 17.33315368558453373012704905927

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