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Properties

Label 2-2349-261.173-c0-0-6
Degree $2$
Conductor $2349$
Sign $0.342 + 0.939i$
Analytic cond. $1.17230$
Root an. cond. $1.08272$
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  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s − 17-s + (−0.499 − 0.866i)22-s + (−0.5 + 0.866i)25-s + 0.999·26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)34-s + (−1 − 1.73i)41-s + (0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)7-s + 8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)13-s + (−0.499 − 0.866i)14-s + (0.5 − 0.866i)16-s − 17-s + (−0.499 − 0.866i)22-s + (−0.5 + 0.866i)25-s + 0.999·26-s + (−0.5 + 0.866i)29-s + (−0.5 + 0.866i)34-s + (−1 − 1.73i)41-s + (0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(1.17230\)
Root analytic conductor: \(1.08272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2349,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.864225862\)
\(L(\frac12)\) \(\approx\) \(1.864225862\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.949236574575763489242640537746, −8.424421247558174261973734677595, −7.27472621269047602996597487632, −6.92405789528774558144501641152, −5.74281376078611196497433155739, −4.74079935359141024773329738533, −3.90853517882023714021971010320, −3.51207808181038410569330838312, −2.16479665743735331232219848984, −1.30889049399874421084846846450, 1.59736743448486564007680171172, 2.53596855720886573411880936667, 4.00553614865281672756648527803, 4.71569927051995902317305188021, 5.47394030218173697574716247415, 6.22840990604063261751098207621, 6.76710318971175644770152157904, 7.82288122298197313375547977668, 8.245496835502830298412590598696, 9.230651093887423964095041606055

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