Properties

Label 1-6025-6025.2488-r0-0-0
Degree $1$
Conductor $6025$
Sign $0.248 + 0.968i$
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  + (−0.544 − 0.838i)2-s + (−0.777 + 0.629i)3-s + (−0.406 + 0.913i)4-s + (0.951 + 0.309i)6-s + (−0.999 + 0.0261i)7-s + (0.987 − 0.156i)8-s + (0.207 − 0.978i)9-s + (0.566 − 0.824i)11-s + (−0.258 − 0.965i)12-s + (0.793 + 0.608i)13-s + (0.566 + 0.824i)14-s + (−0.669 − 0.743i)16-s + (0.649 + 0.760i)17-s + (−0.933 + 0.358i)18-s + (−0.999 + 0.0261i)19-s + ⋯
L(s)  = 1  + (−0.544 − 0.838i)2-s + (−0.777 + 0.629i)3-s + (−0.406 + 0.913i)4-s + (0.951 + 0.309i)6-s + (−0.999 + 0.0261i)7-s + (0.987 − 0.156i)8-s + (0.207 − 0.978i)9-s + (0.566 − 0.824i)11-s + (−0.258 − 0.965i)12-s + (0.793 + 0.608i)13-s + (0.566 + 0.824i)14-s + (−0.669 − 0.743i)16-s + (0.649 + 0.760i)17-s + (−0.933 + 0.358i)18-s + (−0.999 + 0.0261i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.248 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6025\)    =    \(5^{2} \cdot 241\)
Sign: $0.248 + 0.968i$
Analytic conductor: \(27.9799\)
Root analytic conductor: \(27.9799\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6025} (2488, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6025,\ (0:\ ),\ 0.248 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4804252067 + 0.3726736115i\)
\(L(\frac12)\) \(\approx\) \(0.4804252067 + 0.3726736115i\)
\(L(1)\) \(\approx\) \(0.5769671318 - 0.03632446733i\)
\(L(1)\) \(\approx\) \(0.5769671318 - 0.03632446733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.544 - 0.838i)T \)
3 \( 1 + (-0.777 + 0.629i)T \)
7 \( 1 + (-0.999 + 0.0261i)T \)
11 \( 1 + (0.566 - 0.824i)T \)
13 \( 1 + (0.793 + 0.608i)T \)
17 \( 1 + (0.649 + 0.760i)T \)
19 \( 1 + (-0.999 + 0.0261i)T \)
23 \( 1 + (0.972 - 0.233i)T \)
29 \( 1 + (0.0523 - 0.998i)T \)
31 \( 1 + (0.182 + 0.983i)T \)
37 \( 1 + (0.999 - 0.0261i)T \)
41 \( 1 + (-0.156 + 0.987i)T \)
43 \( 1 + (-0.852 - 0.522i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (-0.838 - 0.544i)T \)
59 \( 1 + (-0.358 + 0.933i)T \)
61 \( 1 + (-0.987 - 0.156i)T \)
67 \( 1 + (0.544 + 0.838i)T \)
71 \( 1 + (-0.430 + 0.902i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.891 + 0.453i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.182 - 0.983i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−17.4118096480883657094653924871, −16.925961463758885839431546685695, −16.48135817776624864959782068873, −15.72770204664716419877200542422, −15.1710322122394489703616143942, −14.40702187573821434864345072055, −13.49367484156266313535531502171, −13.08280263374401680808805952132, −12.424535565381385986792128947798, −11.62197106437801029848364429223, −10.72669792944801988322824410887, −10.33492010760035513524310702571, −9.4298305155061220692033545427, −8.98206397145660577210559142101, −7.92371719748781892618915069374, −7.44969031552515581871991387892, −6.57858687544704750762854382851, −6.42258086505163912873495692179, −5.54961193001172196828337294578, −4.92143682552466548267607971825, −4.06443541079883587844871447086, −3.00593461950424520817056201285, −1.91021215641923973843352938628, −1.107926627617869665934673562736, −0.30348508788163367279833625081, 0.85171954199974864098034121127, 1.47611797982559423933992363722, 2.78607888828578729772809726511, 3.40078885504628993028388536785, 4.02782756402533594574653831273, 4.598250713802266743619776438070, 5.78011800574427861595336160027, 6.35049078372806137447997210168, 6.89066258454130963302444781273, 8.10749045784398166360509684967, 8.806830859689101245367266405422, 9.272265460648278425269739449352, 10.02478684418271371667073300000, 10.55982349278949474397419512642, 11.19586896849218508990249869892, 11.700626637159102319024165700380, 12.487903060907067317768466840269, 12.97688899160747036789858298967, 13.70328817159199696953642004184, 14.60257943916632379373839166274, 15.45278029370131700094402795049, 16.213358623024339252026406089251, 16.73225414515255415946673365027, 16.99549702402168491558394228925, 17.823681116463844960345763719951

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