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Properties

Label 2-2349-261.157-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.323192723\)
\(L(\frac12)\) \(\approx\) \(1.323192723\)
\(L(1)\) 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

−9.069335123120073769389743190376, −8.320817880354059959959242162004, −7.78293683741057189996181114854, −7.27626293189250015935945131050, −5.93546036981903555582444391604, −5.62616982008485993670297081354, −4.25578038731871571492729780601, −3.65425987571402826367279226398, −2.53278132018889839526612148136, −1.54269653190541188291018196168, 0.962979849617603967068154075159, 2.27407989590122519820945774080, 3.32188333961569324777361774966, 4.11353682199998354706713951789, 5.17903195802927040404026262782, 5.96520603499672822803406770852, 7.13472242588666156662960619248, 7.30435381472095295935712608177, 7.973123929188648313723052662104, 9.113019579358355156555380466000

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