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Properties

Label 1-51e2-2601.1327-r0-0-0
Degree $1$
Conductor $2601$
Sign $-0.414 - 0.910i$
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  + (−0.998 − 0.0615i)2-s + (0.992 + 0.122i)4-s + (0.650 + 0.759i)5-s + (−0.952 + 0.303i)7-s + (−0.982 − 0.183i)8-s + (−0.602 − 0.798i)10-s + (−0.389 − 0.920i)11-s + (0.332 + 0.943i)13-s + (0.969 − 0.243i)14-s + (0.969 + 0.243i)16-s + (0.445 − 0.895i)19-s + (0.552 + 0.833i)20-s + (0.332 + 0.943i)22-s + (0.213 − 0.976i)23-s + (−0.153 + 0.988i)25-s + (−0.273 − 0.961i)26-s + ⋯
L(s)  = 1  + (−0.998 − 0.0615i)2-s + (0.992 + 0.122i)4-s + (0.650 + 0.759i)5-s + (−0.952 + 0.303i)7-s + (−0.982 − 0.183i)8-s + (−0.602 − 0.798i)10-s + (−0.389 − 0.920i)11-s + (0.332 + 0.943i)13-s + (0.969 − 0.243i)14-s + (0.969 + 0.243i)16-s + (0.445 − 0.895i)19-s + (0.552 + 0.833i)20-s + (0.332 + 0.943i)22-s + (0.213 − 0.976i)23-s + (−0.153 + 0.988i)25-s + (−0.273 − 0.961i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $-0.414 - 0.910i$
Analytic conductor: \(12.0789\)
Root analytic conductor: \(12.0789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (1327, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2601,\ (0:\ ),\ -0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1989310779 - 0.3090575854i\)
\(L(\frac12)\) \(\approx\) \(0.1989310779 - 0.3090575854i\)
\(L(1)\) \(\approx\) \(0.6081364325 + 0.01619009255i\)
\(L(1)\) \(\approx\) \(0.6081364325 + 0.01619009255i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.998 - 0.0615i)T \)
5 \( 1 + (0.650 + 0.759i)T \)
7 \( 1 + (-0.952 + 0.303i)T \)
11 \( 1 + (-0.389 - 0.920i)T \)
13 \( 1 + (0.332 + 0.943i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (0.213 - 0.976i)T \)
29 \( 1 + (-0.389 - 0.920i)T \)
31 \( 1 + (0.332 + 0.943i)T \)
37 \( 1 + (-0.850 - 0.526i)T \)
41 \( 1 + (-0.153 - 0.988i)T \)
43 \( 1 + (-0.696 - 0.717i)T \)
47 \( 1 + (0.213 + 0.976i)T \)
53 \( 1 + (0.739 + 0.673i)T \)
59 \( 1 + (-0.908 + 0.417i)T \)
61 \( 1 + (-0.908 - 0.417i)T \)
67 \( 1 + (-0.998 + 0.0615i)T \)
71 \( 1 + (0.739 + 0.673i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (0.552 + 0.833i)T \)
83 \( 1 + (-0.153 + 0.988i)T \)
89 \( 1 + (-0.982 + 0.183i)T \)
97 \( 1 + (-0.952 + 0.303i)T \)
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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

−19.80483795141207024374673584283, −18.71012460506661963620045552028, −18.12644016034199001627981705299, −17.50422673106101599030204163988, −16.75148606080528982600480919981, −16.33187832700556777808391599784, −15.48054248187653735768275104129, −14.96276804114655269193595129783, −13.7216518416235961270411355980, −13.053479547255835235195447116451, −12.4535961119129216993728209573, −11.69878313518574523009682741491, −10.56085064061614767015693321693, −9.99509562452858466310560399977, −9.61039133208447828946888857542, −8.78730965537708241667500597486, −7.944884454181820938825702431784, −7.314476565546007840677177574814, −6.389814030072425254158411677923, −5.709414839319138159781918982236, −4.97436324297107956575300626530, −3.6218796285998369432421827361, −2.8744019389203282203917372740, −1.78958303606252410489903624496, −1.11293107859535474679293766850, 0.168346711832209409276770045761, 1.45315063566986738691764736347, 2.52421103572997396329974674174, 2.92293827441955936292059128271, 3.85445529076243764373594582051, 5.390035633761295539496200511992, 6.11650406001320488555242899783, 6.726851504030510257872805445254, 7.26434421091450254985235078204, 8.44459448350048326026323152648, 9.085319952677645196039559780676, 9.593416709967109576892023561369, 10.56798549162674684217883864419, 10.8635787225317693171455949829, 11.80805514316658380909102864451, 12.54276059564809430911981652810, 13.63422721304041241765584456977, 13.98005151308916839245171607684, 15.228440953020582377595772972813, 15.656031250116175180053947686858, 16.50209747451215738504759729953, 16.97116636654517270786423941185, 17.93729136106820572502506623157, 18.46971906385762068154265160022, 19.118299278236477872149131345431

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