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Properties

Label 1-2349-2349.1984-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.624 + 0.780i$
Analytic cond. $252.435$
Root an. cond. $252.435$
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  + (−0.116 − 0.993i)2-s + (−0.973 + 0.230i)4-s + (−0.893 − 0.448i)5-s + (−0.286 − 0.957i)7-s + (0.342 + 0.939i)8-s + (−0.342 + 0.939i)10-s + (−0.998 − 0.0581i)11-s + (−0.597 + 0.802i)13-s + (−0.918 + 0.396i)14-s + (0.893 − 0.448i)16-s + (0.642 + 0.766i)17-s + (−0.642 + 0.766i)19-s + (0.973 + 0.230i)20-s + (0.0581 + 0.998i)22-s + (−0.286 + 0.957i)23-s + ⋯
L(s)  = 1  + (−0.116 − 0.993i)2-s + (−0.973 + 0.230i)4-s + (−0.893 − 0.448i)5-s + (−0.286 − 0.957i)7-s + (0.342 + 0.939i)8-s + (−0.342 + 0.939i)10-s + (−0.998 − 0.0581i)11-s + (−0.597 + 0.802i)13-s + (−0.918 + 0.396i)14-s + (0.893 − 0.448i)16-s + (0.642 + 0.766i)17-s + (−0.642 + 0.766i)19-s + (0.973 + 0.230i)20-s + (0.0581 + 0.998i)22-s + (−0.286 + 0.957i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1984, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.624 + 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02158501650 - 0.04492907318i\)
\(L(\frac12)\) \(\approx\) \(0.02158501650 - 0.04492907318i\)
\(L(1)\) \(\approx\) \(0.5474463053 - 0.2916530611i\)
\(L(1)\) \(\approx\) \(0.5474463053 - 0.2916530611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.116 - 0.993i)T \)
5 \( 1 + (-0.893 - 0.448i)T \)
7 \( 1 + (-0.286 - 0.957i)T \)
11 \( 1 + (-0.998 - 0.0581i)T \)
13 \( 1 + (-0.597 + 0.802i)T \)
17 \( 1 + (0.642 + 0.766i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.286 + 0.957i)T \)
31 \( 1 + (0.727 - 0.686i)T \)
37 \( 1 + (0.984 - 0.173i)T \)
41 \( 1 + (-0.116 + 0.993i)T \)
43 \( 1 + (0.549 - 0.835i)T \)
47 \( 1 + (0.727 + 0.686i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.0581 - 0.998i)T \)
61 \( 1 + (-0.230 + 0.973i)T \)
67 \( 1 + (-0.396 - 0.918i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.342 + 0.939i)T \)
79 \( 1 + (-0.116 - 0.993i)T \)
83 \( 1 + (-0.993 + 0.116i)T \)
89 \( 1 + (-0.342 - 0.939i)T \)
97 \( 1 + (-0.448 - 0.893i)T \)
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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

−19.572872098857852760277957947977, −18.97755621616451310383271680346, −18.324056142149721251471708901246, −17.90436886950991836358715798624, −16.861421099644381751058323317725, −16.0754978244065514393234019647, −15.52519093281361529616643776944, −15.11121502084436964015576062710, −14.40136295911146251441775403650, −13.500457498280994789427677579695, −12.49321103039735533574681339244, −12.2750236002624763249673273336, −10.99362780058919579751962677991, −10.28732215658970516575584485520, −9.50166930263914229365007103034, −8.55826893496148995315121907871, −8.02461843304173533578139751690, −7.33116204793781170273562038129, −6.5865656195452780705401529247, −5.698943614952815978385101398486, −4.99009186100913193169840011876, −4.27199725457765713640894945446, −2.99982826603419984305641623610, −2.572048496266771358111578087773, −0.67449058627163487816207335841, 0.015867373143736455065050575708, 0.92809575814560202136267711903, 1.85605297154206894983283069956, 2.93588881224830015629803214132, 3.83253623761079654412537663454, 4.27033455121391724733016100061, 5.09920783622091392452642893568, 6.156689949232366119969259118796, 7.54736953103652495530512806441, 7.80813818515989082355095152955, 8.64729716593851405257533273311, 9.70564223799273280623854157295, 10.14130571888193848876185178441, 11.02872977438730620841128844411, 11.58736388111902987261591406458, 12.52616221116097458195242584686, 12.8644003664482040524181205416, 13.73913550160588146593670751782, 14.45977842767108110450982146655, 15.347622491013106725810725624736, 16.32288653063324285045937083417, 16.88848733049597878777102716620, 17.45991434261444950835735332233, 18.61945142681396307951606904504, 19.0892474340443160880829448544

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