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Properties

Label 2-2349-261.173-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9486758371\)
\(L(\frac12)\) \(\approx\) \(0.9486758371\)
\(L(1)\) 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

−9.446295129169520255240476262996, −8.320865341611171081410594024599, −7.79390817150058365486026692100, −7.28923919052202620910187721647, −6.51022253449871771757361718452, −5.71507915810527708854598076809, −4.67572823611261978021256647948, −3.86887968909074910615004539819, −2.73437569766984234489949647826, −1.37307258523795463114404955882, 0.837912564029636181169903870958, 2.06882255985564483004333347178, 2.91310204240849799603633423801, 3.68815216195576276136331921070, 5.26206935524964163026334869826, 5.63913720750977585500279247924, 6.45021212996510603166879184992, 7.74838802162324416450205172911, 8.412138299847160747349755640002, 8.910984258929684722708219687622

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