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Properties

Label 2-2349-261.86-c0-0-2
Degree $2$
Conductor $2349$
Sign $0.984 - 0.173i$
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  + (0.173 + 0.300i)2-s + (0.439 − 0.761i)4-s + (−0.173 − 0.300i)7-s + 0.652·8-s + (0.766 + 1.32i)11-s + (−0.766 + 1.32i)13-s + (0.0603 − 0.104i)14-s + (−0.326 − 0.565i)16-s + 1.87·17-s + (−0.266 + 0.460i)22-s + (−0.5 − 0.866i)25-s − 0.532·26-s − 0.305·28-s + (0.5 + 0.866i)29-s + (0.439 − 0.761i)32-s + ⋯
L(s)  = 1  + (0.173 + 0.300i)2-s + (0.439 − 0.761i)4-s + (−0.173 − 0.300i)7-s + 0.652·8-s + (0.766 + 1.32i)11-s + (−0.766 + 1.32i)13-s + (0.0603 − 0.104i)14-s + (−0.326 − 0.565i)16-s + 1.87·17-s + (−0.266 + 0.460i)22-s + (−0.5 − 0.866i)25-s − 0.532·26-s − 0.305·28-s + (0.5 + 0.866i)29-s + (0.439 − 0.761i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2349\)    =    \(3^{4} \cdot 29\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(1.17230\)
Root analytic conductor: \(1.08272\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2349} (782, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2349,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.552304135\)
\(L(\frac12)\) \(\approx\) \(1.552304135\)
\(L(1)\) 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 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.87T + T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
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

−9.574662895076812921844397251110, −8.323422910684662018532065596855, −7.31338095099300864898628147474, −6.91452978289678888242878716852, −6.24352678200498750626368114292, −5.16520161464829599761571265908, −4.59919316881294679288271569022, −3.61133162486075192485793032145, −2.20364055305636040754759168336, −1.39374677912564903257679381272, 1.22261946604889817641663463973, 2.74584364361073270978424699473, 3.24201053608368482030496591714, 4.03410183034713769146601560649, 5.37014638333576410440864263689, 5.89239159347669700491338097888, 6.90035266086346253877236794232, 7.891277851768720449309637302977, 8.072671846802172619397432903018, 9.154447982151124951211517018420

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