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Properties

Label 2-2349-261.86-c0-0-1
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  + (0.939 + 1.62i)2-s + (−1.26 + 2.19i)4-s + (0.939 + 1.62i)7-s − 2.87·8-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)13-s + (−1.76 + 3.05i)14-s + (−1.43 − 2.49i)16-s + 1.53·17-s + (0.326 − 0.565i)22-s + (−0.5 − 0.866i)25-s − 0.652·26-s − 4.75·28-s + (−0.5 − 0.866i)29-s + (1.26 − 2.19i)32-s + ⋯
L(s)  = 1  + (0.939 + 1.62i)2-s + (−1.26 + 2.19i)4-s + (0.939 + 1.62i)7-s − 2.87·8-s + (−0.173 − 0.300i)11-s + (−0.173 + 0.300i)13-s + (−1.76 + 3.05i)14-s + (−1.43 − 2.49i)16-s + 1.53·17-s + (0.326 − 0.565i)22-s + (−0.5 − 0.866i)25-s − 0.652·26-s − 4.75·28-s + (−0.5 − 0.866i)29-s + (1.26 − 2.19i)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

\(L(\frac{1}{2})\) \(\approx\) \(1.893324320\)
\(L(\frac12)\) \(\approx\) \(1.893324320\)
\(L(1)\) 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 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
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 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 0.347T + T^{2} \)
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

−9.106034900812612965976127045923, −8.451791944245686735614044908972, −7.955230117831130385058537795384, −7.28346040436681689910829719714, −6.15889656710566183989168694266, −5.68924531187133946454840636212, −5.16026954037864162177387723111, −4.31287182914823307544266950779, −3.28830166246203580413371851445, −2.21039186434728909219936165965, 1.05791711734422744677005334687, 1.76017716341023010294763205871, 3.15196674231519060306929874599, 3.71589865119563659722596788210, 4.64118465694282537174484254730, 5.10535188266800588153173626536, 6.06356792026455164812294320797, 7.39438284185195053924571042968, 7.88013013745734756998114814827, 9.171985405594236177287930061232

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