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Properties

Label 1-6025-6025.1578-r0-0-0
Degree $1$
Conductor $6025$
Sign $-0.367 + 0.929i$
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  + (0.933 + 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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (1578, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ -0.367 + 0.929i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.933 + 0.358i)T \)
3 \( 1 + (-0.838 - 0.544i)T \)
7 \( 1 + (0.333 + 0.942i)T \)
11 \( 1 + (-0.0261 - 0.999i)T \)
13 \( 1 + (-0.793 - 0.608i)T \)
17 \( 1 + (0.972 + 0.233i)T \)
19 \( 1 + (0.333 + 0.942i)T \)
23 \( 1 + (-0.0784 + 0.996i)T \)
29 \( 1 + (-0.629 + 0.777i)T \)
31 \( 1 + (0.725 + 0.688i)T \)
37 \( 1 + (-0.333 - 0.942i)T \)
41 \( 1 + (0.891 + 0.453i)T \)
43 \( 1 + (0.760 - 0.649i)T \)
47 \( 1 + (0.156 - 0.987i)T \)
53 \( 1 + (0.358 + 0.933i)T \)
59 \( 1 + (-0.998 - 0.0523i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (-0.933 - 0.358i)T \)
71 \( 1 + (-0.878 + 0.477i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (-0.987 + 0.156i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.725 - 0.688i)T \)
97 \( 1 + (-0.913 - 0.406i)T \)
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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

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

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