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Properties

Label 2-345-345.29-c0-0-0
Degree $2$
Conductor $345$
Sign $0.392 - 0.919i$
Analytic cond. $0.172177$
Root an. cond. $0.414942$
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  + (−0.797 + 1.74i)2-s + (−0.959 − 0.281i)3-s + (−1.75 − 2.02i)4-s + (0.841 − 0.540i)5-s + (1.25 − 1.45i)6-s + (3.09 − 0.909i)8-s + (0.841 + 0.540i)9-s + (0.273 + 1.89i)10-s + (1.11 + 2.44i)12-s + (−0.959 + 0.281i)15-s + (−0.500 + 3.47i)16-s + (0.857 − 0.989i)17-s + (−1.61 + 1.03i)18-s + (0.186 + 0.215i)19-s + (−2.57 − 0.755i)20-s + ⋯
L(s)  = 1  + (−0.797 + 1.74i)2-s + (−0.959 − 0.281i)3-s + (−1.75 − 2.02i)4-s + (0.841 − 0.540i)5-s + (1.25 − 1.45i)6-s + (3.09 − 0.909i)8-s + (0.841 + 0.540i)9-s + (0.273 + 1.89i)10-s + (1.11 + 2.44i)12-s + (−0.959 + 0.281i)15-s + (−0.500 + 3.47i)16-s + (0.857 − 0.989i)17-s + (−1.61 + 1.03i)18-s + (0.186 + 0.215i)19-s + (−2.57 − 0.755i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(345\)    =    \(3 \cdot 5 \cdot 23\)
Sign: $0.392 - 0.919i$
Analytic conductor: \(0.172177\)
Root analytic conductor: \(0.414942\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{345} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 345,\ (\ :0),\ 0.392 - 0.919i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4441961962\)
\(L(\frac12)\) \(\approx\) \(0.4441961962\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.959 + 0.281i)T \)
5 \( 1 + (-0.841 + 0.540i)T \)
23 \( 1 + (-0.415 - 0.909i)T \)
good2 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
7 \( 1 + (0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
29 \( 1 + (0.142 + 0.989i)T^{2} \)
31 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
37 \( 1 + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.841 - 0.540i)T^{2} \)
47 \( 1 - 0.830T + T^{2} \)
53 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.239 + 1.66i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.841 - 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−11.91089065759182426988354114824, −10.54961406989724961272142724406, −9.764595797492933284693129556603, −9.096118387094815778649703111029, −7.87399618298587003716072794061, −7.12876864929447214911294639144, −6.11542446581340811521417616236, −5.46272047483734989227850091498, −4.70853200876993286001993447405, −1.23386469457785985176323931721, 1.46459226336072092376792273284, 2.93522196579340408021302567228, 4.17102270130317430548406875996, 5.45391537050398861032702333170, 6.81838041435685354096406553739, 8.188313847907576118045191991167, 9.366145756786886366382605886330, 10.00029626495290873643209817560, 10.68446215545145526478523557791, 11.21942454144155133550407183043

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