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Properties

Label 2-2349-261.157-c0-0-4
Degree $2$
Conductor $2349$
Sign $0.630 - 0.776i$
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  + (0.866 + 0.5i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)7-s + (0.499 + 0.866i)16-s + (−1 − i)17-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)23-s + 0.999i·28-s + (0.5 + 0.866i)29-s + (−1.36 + 0.366i)31-s + 0.999i·35-s + (1 − i)37-s + (−0.366 − 1.36i)41-s + (−1.36 − 0.366i)47-s − 53-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)4-s + (0.866 + 0.5i)5-s + (0.5 + 0.866i)7-s + (0.499 + 0.866i)16-s + (−1 − i)17-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)23-s + 0.999i·28-s + (0.5 + 0.866i)29-s + (−1.36 + 0.366i)31-s + 0.999i·35-s + (1 − i)37-s + (−0.366 − 1.36i)41-s + (−1.36 − 0.366i)47-s − 53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(1.17230\)
Root analytic conductor: \(1.08272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (1810, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2349,\ (\ :0),\ 0.630 - 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.752513612\)
\(L(\frac12)\) \(\approx\) \(1.752513612\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.866 + 0.5i)T^{2} \)
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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

−9.037616296471888635840913829636, −8.712756570888152038641522784046, −7.58492059612502583379036852827, −6.93541403122397519785171848703, −6.25619627954591655820484970627, −5.49654833368979946218486382615, −4.58774898356782282150440337824, −3.24060903095444923943417211503, −2.44891492967330118606564591510, −1.85845618377890606085168874854, 1.35204363059341798769391924494, 1.92150647527524357003169667662, 3.18988805440586236504421122481, 4.41437258879506967575592719168, 5.14162138046082510947805721888, 6.10833764425395088482163905109, 6.53349405948757499173598083347, 7.56948923340759331375661129674, 8.140485503715759833526620191175, 9.294436595114122270620692441260

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