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Properties

Label 1-2349-2349.1051-r0-0-0
Degree $1$
Conductor $2349$
Sign $0.965 - 0.261i$
Analytic cond. $10.9087$
Root an. cond. $10.9087$
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  + (0.945 − 0.326i)2-s + (0.786 − 0.616i)4-s + (−0.610 + 0.792i)5-s + (−0.927 + 0.373i)7-s + (0.542 − 0.840i)8-s + (−0.318 + 0.947i)10-s + (−0.960 + 0.278i)11-s + (0.649 − 0.760i)13-s + (−0.755 + 0.655i)14-s + (0.238 − 0.971i)16-s + (0.766 + 0.642i)17-s + (−0.0249 − 0.999i)19-s + (0.00831 + 0.999i)20-s + (−0.816 + 0.576i)22-s + (−0.157 + 0.987i)23-s + ⋯
L(s)  = 1  + (0.945 − 0.326i)2-s + (0.786 − 0.616i)4-s + (−0.610 + 0.792i)5-s + (−0.927 + 0.373i)7-s + (0.542 − 0.840i)8-s + (−0.318 + 0.947i)10-s + (−0.960 + 0.278i)11-s + (0.649 − 0.760i)13-s + (−0.755 + 0.655i)14-s + (0.238 − 0.971i)16-s + (0.766 + 0.642i)17-s + (−0.0249 − 0.999i)19-s + (0.00831 + 0.999i)20-s + (−0.816 + 0.576i)22-s + (−0.157 + 0.987i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(10.9087\)
Root analytic conductor: \(10.9087\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (0:\ ),\ 0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.313525574 - 0.3075183902i\)
\(L(\frac12)\) \(\approx\) \(2.313525574 - 0.3075183902i\)
\(L(1)\) \(\approx\) \(1.536365605 - 0.1644703431i\)
\(L(1)\) \(\approx\) \(1.536365605 - 0.1644703431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.945 - 0.326i)T \)
5 \( 1 + (-0.610 + 0.792i)T \)
7 \( 1 + (-0.927 + 0.373i)T \)
11 \( 1 + (-0.960 + 0.278i)T \)
13 \( 1 + (0.649 - 0.760i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (-0.0249 - 0.999i)T \)
23 \( 1 + (-0.157 + 0.987i)T \)
31 \( 1 + (0.302 - 0.953i)T \)
37 \( 1 + (0.921 + 0.388i)T \)
41 \( 1 + (-0.993 + 0.116i)T \)
43 \( 1 + (0.991 - 0.132i)T \)
47 \( 1 + (-0.556 + 0.830i)T \)
53 \( 1 + (0.0747 - 0.997i)T \)
59 \( 1 + (-0.0581 + 0.998i)T \)
61 \( 1 + (0.786 + 0.616i)T \)
67 \( 1 + (0.807 + 0.590i)T \)
71 \( 1 + (0.542 + 0.840i)T \)
73 \( 1 + (0.698 + 0.715i)T \)
79 \( 1 + (0.334 - 0.942i)T \)
83 \( 1 + (-0.710 + 0.704i)T \)
89 \( 1 + (0.995 - 0.0995i)T \)
97 \( 1 + (0.908 - 0.418i)T \)
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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

−19.85935001907174040338274686923, −18.94690564123390207319237251070, −18.35861708264117477822427762752, −16.968174400289691813605828993345, −16.51319313514698686352438668135, −16.016766114482185837436308770164, −15.56688327522254316331089704818, −14.42467848154032846596115037813, −13.83289003293706930511714888753, −13.09118004629135831682955059608, −12.48578700583546041530459891086, −11.963565866497613658304501659538, −11.02906600811887590701089961602, −10.280862897698530875086906630492, −9.26269468668759130296424542421, −8.2852390344104110613623363057, −7.78297171452813580305134167396, −6.86382605916848935562503346879, −6.13635525877927227532048272741, −5.29915532331066493356567398606, −4.569862759249195481823853182522, −3.7022295254280343761763352297, −3.20655065651491774834328592100, −2.06715845026012837404155878410, −0.78394713449031432854536533010, 0.77115393425863680005428418761, 2.23541307797545316699934485225, 2.9396139966974443063917121549, 3.47143462288043879496251060089, 4.29505556886599119612413070000, 5.432038110752315491612301954083, 5.963509928055996493708188838148, 6.79200374280347753692615882912, 7.55093810360260405954654791967, 8.30141571678419615059602711005, 9.71870655666426828619914108679, 10.1555384171942337893282118545, 11.004875015243154780234880486726, 11.56033041707365700766803253947, 12.44793472045449712655697579531, 13.103276205147998150157400706956, 13.54872505860300475127930475005, 14.67208730383214809811533802371, 15.267234652536100218739542589806, 15.692635130044432171064114732372, 16.28138190804650305115562679220, 17.50044195364592499787045692071, 18.44904006143186320066746323341, 18.96416393612484146736018279487, 19.63081737662386716427525979286

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