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Properties

Label 1-2349-2349.592-r1-0-0
Degree $1$
Conductor $2349$
Sign $-0.290 - 0.956i$
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  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.342 + 0.939i)8-s + (0.342 + 0.939i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)13-s + (−0.116 − 0.993i)14-s + (−0.835 + 0.549i)16-s + (−0.642 + 0.766i)17-s + (0.642 + 0.766i)19-s + (−0.286 + 0.957i)20-s + (−0.893 + 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯
L(s)  = 1  + (0.802 + 0.597i)2-s + (0.286 + 0.957i)4-s + (0.835 + 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.342 + 0.939i)8-s + (0.342 + 0.939i)10-s + (−0.448 + 0.893i)11-s + (−0.396 + 0.918i)13-s + (−0.116 − 0.993i)14-s + (−0.835 + 0.549i)16-s + (−0.642 + 0.766i)17-s + (0.642 + 0.766i)19-s + (−0.286 + 0.957i)20-s + (−0.893 + 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.290 - 0.956i$
Analytic conductor: \(252.435\)
Root analytic conductor: \(252.435\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (592, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2349,\ (1:\ ),\ -0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.145160031 + 1.544127939i\)
\(L(\frac12)\) \(\approx\) \(-1.145160031 + 1.544127939i\)
\(L(1)\) \(\approx\) \(1.062205324 + 1.015931365i\)
\(L(1)\) \(\approx\) \(1.062205324 + 1.015931365i\)

Euler product

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

−19.08518831096293000612963071056, −18.19463522608656710741102524030, −17.7660045397811983821178360831, −16.49106565322627241301645965057, −15.93108420690683196723180036770, −15.36088016600005064414519258765, −14.34511173397089185405667641959, −13.67980885399524723381308862035, −13.05126458327311090826259158920, −12.60495164557044768823333609365, −11.76556583636495929239972022659, −10.9626852173562796115117378765, −10.16128002017409468522828994561, −9.44934023249872093755852093529, −8.9302097893313265021644585452, −7.813839345879405572298555235517, −6.575193063033801270236185627323, −5.9861529152981941463548136140, −5.30817091540115664912301135742, −4.757324701252823466398302778437, −3.54174338032476567770716742772, −2.59549722135111541066909384495, −2.36503223893036213224400615540, −0.90818747784813422902628873413, −0.23925740207061322051292369596, 1.68966637368860584001884501910, 2.32821203875113254785842346271, 3.41810608895325276884967234297, 4.02306168595398435305906948524, 4.98140931937332516681901228589, 5.79243821264219503571233310542, 6.552407352806057689439060579133, 7.12722965091367942297158148714, 7.70561051016040861547825523851, 8.93267706855543568004201274964, 9.722360495799982795393443927245, 10.411243040430369579658801235065, 11.205047947813851603875756307591, 12.34388874068727747790988350284, 12.70894108782567058085208157085, 13.77051160978162789170856487144, 13.96389904393225850764947877295, 14.753666739530541032330033400781, 15.63556543607278529101476785224, 16.18714715882599238970878420526, 17.1295675158417793091537910771, 17.49517599376885583153905663164, 18.26374667489915450792544541239, 19.26766279002254039169680909199, 20.07299458890148109409582086625

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