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Properties

Label 1-51e2-2601.25-r0-0-0
Degree $1$
Conductor $2601$
Sign $0.508 - 0.861i$
Analytic cond. $12.0789$
Root an. cond. $12.0789$
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  + (−0.833 − 0.552i)2-s + (0.389 + 0.920i)4-s + (0.901 − 0.431i)5-s + (−0.539 − 0.842i)7-s + (0.183 − 0.982i)8-s + (−0.990 − 0.138i)10-s + (−0.788 + 0.614i)11-s + (−0.650 + 0.759i)13-s + (−0.0153 + 0.999i)14-s + (−0.696 + 0.717i)16-s + (−0.895 − 0.445i)19-s + (0.749 + 0.662i)20-s + (0.997 − 0.0769i)22-s + (0.459 − 0.888i)23-s + (0.626 − 0.779i)25-s + (0.961 − 0.273i)26-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (0.389 + 0.920i)4-s + (0.901 − 0.431i)5-s + (−0.539 − 0.842i)7-s + (0.183 − 0.982i)8-s + (−0.990 − 0.138i)10-s + (−0.788 + 0.614i)11-s + (−0.650 + 0.759i)13-s + (−0.0153 + 0.999i)14-s + (−0.696 + 0.717i)16-s + (−0.895 − 0.445i)19-s + (0.749 + 0.662i)20-s + (0.997 − 0.0769i)22-s + (0.459 − 0.888i)23-s + (0.626 − 0.779i)25-s + (0.961 − 0.273i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.508 - 0.861i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ 0.508 - 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8337840618 - 0.4761962008i\)
\(L(\frac12)\) \(\approx\) \(0.8337840618 - 0.4761962008i\)
\(L(1)\) \(\approx\) \(0.7007435988 - 0.2269732226i\)
\(L(1)\) \(\approx\) \(0.7007435988 - 0.2269732226i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.833 - 0.552i)T \)
5 \( 1 + (0.901 - 0.431i)T \)
7 \( 1 + (-0.539 - 0.842i)T \)
11 \( 1 + (-0.788 + 0.614i)T \)
13 \( 1 + (-0.650 + 0.759i)T \)
19 \( 1 + (-0.895 - 0.445i)T \)
23 \( 1 + (0.459 - 0.888i)T \)
29 \( 1 + (0.614 + 0.788i)T \)
31 \( 1 + (-0.997 + 0.0769i)T \)
37 \( 1 + (0.228 + 0.973i)T \)
41 \( 1 + (-0.107 + 0.994i)T \)
43 \( 1 + (0.243 + 0.969i)T \)
47 \( 1 + (0.952 + 0.303i)T \)
53 \( 1 + (0.673 - 0.739i)T \)
59 \( 1 + (-0.577 + 0.816i)T \)
61 \( 1 + (0.168 - 0.985i)T \)
67 \( 1 + (0.552 + 0.833i)T \)
71 \( 1 + (-0.0461 - 0.998i)T \)
73 \( 1 + (0.486 - 0.873i)T \)
79 \( 1 + (0.662 - 0.749i)T \)
83 \( 1 + (-0.626 + 0.779i)T \)
89 \( 1 + (0.982 - 0.183i)T \)
97 \( 1 + (0.842 - 0.539i)T \)
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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

−19.21216072158068189276820089808, −18.62462918887720388039069550116, −18.175111588787945738848819481832, −17.249667765149381856556524476218, −16.94557165405627496920097061823, −15.79907034000905702394317667157, −15.44785261080188643491613292826, −14.68344074011522533191640350797, −13.941470826909876619500095676755, −13.11758797114824111220631437188, −12.39612159062850409469070381564, −11.31167000681447148267809115315, −10.47872117428040929310817191506, −10.13195736694739986013040648952, −9.19908380747536492869531788435, −8.74565164180447673506864717371, −7.75402537540592150377708185050, −7.094864965339514565216284778232, −6.11297855474692450550485395794, −5.65936946537338427977859010017, −5.144229653540384028610653845414, −3.52188583144547209656339138717, −2.413745462218459403072860751405, −2.18080815056480427903399219002, −0.68099713408673455778543122767, 0.59412473899218150915101586883, 1.646259873497935565737269054883, 2.39597090234009161141823133342, 3.13820040280822715437598342846, 4.414035155228559056856540822825, 4.83420522516882982068440917217, 6.28649375629277739598252112157, 6.842390753837317974968560183034, 7.574627838912866352872966024871, 8.53986140493417213521064688816, 9.23990625677110916035420347498, 9.874445468852592526782363817251, 10.44553366361590609646302946874, 11.03423379617062105885339497334, 12.18321671531867364783244528945, 12.84416223520122545699370092848, 13.18953421573525513595539232201, 14.14528527143419946919893857281, 15.02974822838240605783688594195, 16.114136814366300711281003341906, 16.69066906050422286841374642760, 17.08467161373624906447510253831, 17.898091326924605364850721425531, 18.44814572043473381084166855049, 19.33909843033304281405522506587

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