↑

Properties

Label 2-2349-261.157-c0-0-5
Degree $2$
Conductor $2349$
Sign $-0.776 - 0.630i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.866 − 0.5i)4-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−1 + 0.999i)10-s + (−1.36 + 0.366i)14-s + (−0.499 − 0.866i)16-s + (−1 − i)17-s + (1 + i)19-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)23-s + 0.999i·28-s + (−0.866 + 0.5i)29-s + (−1.36 + 0.366i)32-s + (−1.73 + i)34-s + 0.999i·35-s + ⋯
L(s)  = 1  + (0.366 − 1.36i)2-s + (−0.866 − 0.5i)4-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (−1 + 0.999i)10-s + (−1.36 + 0.366i)14-s + (−0.499 − 0.866i)16-s + (−1 − i)17-s + (1 + i)19-s + (0.499 + 0.866i)20-s + (0.5 − 0.866i)23-s + 0.999i·28-s + (−0.866 + 0.5i)29-s + (−1.36 + 0.366i)32-s + (−1.73 + i)34-s + 0.999i·35-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $-0.776 - 0.630i$
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: \((2,\ 2349,\ (\ :0),\ -0.776 - 0.630i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8874163390\)
\(L(\frac12)\) \(\approx\) \(0.8874163390\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
good2 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + (-0.866 - 0.5i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.963956606185261339201570092853, −7.936746814104209464190566611983, −7.24655512191826519492940558876, −6.45829714350991732571217286888, −4.98090535368269701049946964352, −4.52086432453538568732430406329, −3.58298636811914662113846003914, −3.09316332512652184969404004109, −1.73850473420676377137760132437, −0.52403242093895125540826430104, 2.13981867595504848408537893201, 3.36593735107322594558647321970, 4.12841951106615396474764608264, 5.21954986363095026238843576385, 5.76864295279639176127127827323, 6.65787263230301847667854131133, 7.23752689519024410722807635587, 7.78965675411524727968701225764, 8.821625400142158115608634244477, 9.122907446640266811316296524778

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