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Properties

Label 1-2652-2652.839-r0-0-0
Degree $1$
Conductor $2652$
Sign $0.287 + 0.957i$
Analytic cond. $12.3158$
Root an. cond. $12.3158$
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  + (−0.923 + 0.382i)5-s + (0.793 − 0.608i)7-s + (0.991 − 0.130i)11-s + (0.258 + 0.965i)19-s + (0.130 + 0.991i)23-s + (0.707 − 0.707i)25-s + (−0.608 + 0.793i)29-s + (0.382 + 0.923i)31-s + (−0.5 + 0.866i)35-s + (−0.608 + 0.793i)37-s + (−0.130 − 0.991i)41-s + (0.258 + 0.965i)43-s − 47-s + (0.258 − 0.965i)49-s + (−0.707 − 0.707i)53-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)5-s + (0.793 − 0.608i)7-s + (0.991 − 0.130i)11-s + (0.258 + 0.965i)19-s + (0.130 + 0.991i)23-s + (0.707 − 0.707i)25-s + (−0.608 + 0.793i)29-s + (0.382 + 0.923i)31-s + (−0.5 + 0.866i)35-s + (−0.608 + 0.793i)37-s + (−0.130 − 0.991i)41-s + (0.258 + 0.965i)43-s − 47-s + (0.258 − 0.965i)49-s + (−0.707 − 0.707i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2652\)    =    \(2^{2} \cdot 3 \cdot 13 \cdot 17\)
Sign: $0.287 + 0.957i$
Analytic conductor: \(12.3158\)
Root analytic conductor: \(12.3158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2652} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2652,\ (0:\ ),\ 0.287 + 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113376134 + 0.8284810265i\)
\(L(\frac12)\) \(\approx\) \(1.113376134 + 0.8284810265i\)
\(L(1)\) \(\approx\) \(1.010611188 + 0.1515554582i\)
\(L(1)\) \(\approx\) \(1.010611188 + 0.1515554582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (0.793 - 0.608i)T \)
11 \( 1 + (0.991 - 0.130i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.130 + 0.991i)T \)
29 \( 1 + (-0.608 + 0.793i)T \)
31 \( 1 + (0.382 + 0.923i)T \)
37 \( 1 + (-0.608 + 0.793i)T \)
41 \( 1 + (-0.130 - 0.991i)T \)
43 \( 1 + (0.258 + 0.965i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (-0.965 + 0.258i)T \)
61 \( 1 + (0.608 + 0.793i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (-0.991 - 0.130i)T \)
73 \( 1 + (-0.923 + 0.382i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.130 - 0.991i)T \)
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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

−19.07012243655579840984565833151, −18.6957080204017238584403108442, −17.63427592510758332715414285775, −17.1508172989444635046707729644, −16.31553663599306329153448230394, −15.56566794347279820643048464002, −14.95886437105810755834119041062, −14.420352732266932682549292892492, −13.45702203826923383512298696530, −12.58834035223408281148220772300, −11.925879639669497590921325949106, −11.415955537151281437614756000813, −10.81136889467824804632723011640, −9.549698481120331213998304750638, −8.97665715908289616483215791943, −8.277218165910726831852905402109, −7.60809708524111588805023495308, −6.76790921423813740334614719727, −5.889902895059387133818229663294, −4.86585207369074344263132051924, −4.411747079247483117751074959436, −3.53166223913411158799992123287, −2.49735389830782654230144760947, −1.56354283144712270042325852555, −0.49883020773887868657303140350, 1.1022226161260257702573834994, 1.75445156168120872482097602399, 3.253049137776103838474136087, 3.66374619180290592188284313452, 4.52331272322006619011023565255, 5.29631484473248066283085839903, 6.42045205627805599214790107410, 7.1118510218357661416946354217, 7.783849780067586934566429828934, 8.42607778590946510050402681860, 9.28364844049575483988114485816, 10.27416479959832808196345078439, 10.904621016157681605633479612834, 11.675863420457925623298667300382, 12.022123088388557054375433684591, 13.05816775750509746126036056957, 14.063690066765724976044220274253, 14.44346080403100959259430709545, 15.11008126706130395949886556416, 15.993255425697651258206995369156, 16.60490413760209857920379384277, 17.39711926413491231070458690536, 18.00632759995742761831089692214, 18.899500477798929891248296995552, 19.46333339622152092637671343788

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