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Properties

Label 2-2340-12.11-c1-0-1
Degree $2$
Conductor $2340$
Sign $-0.395 - 0.918i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.09 − 0.890i)2-s + (0.415 − 1.95i)4-s − i·5-s + 3.54i·7-s + (−1.28 − 2.51i)8-s + (−0.890 − 1.09i)10-s − 6.08·11-s + 13-s + (3.15 + 3.89i)14-s + (−3.65 − 1.62i)16-s + 3.53i·17-s − 7.05i·19-s + (−1.95 − 0.415i)20-s + (−6.68 + 5.41i)22-s − 8.80·23-s + ⋯
L(s)  = 1  + (0.777 − 0.629i)2-s + (0.207 − 0.978i)4-s − 0.447i·5-s + 1.34i·7-s + (−0.454 − 0.890i)8-s + (−0.281 − 0.347i)10-s − 1.83·11-s + 0.277·13-s + (0.843 + 1.04i)14-s + (−0.913 − 0.406i)16-s + 0.857i·17-s − 1.61i·19-s + (−0.437 − 0.0929i)20-s + (−1.42 + 1.15i)22-s − 1.83·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.395 - 0.918i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1691, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1973427196\)
\(L(\frac12)\) \(\approx\) \(0.1973427196\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.09 + 0.890i)T \)
3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 - T \)
good7 \( 1 - 3.54iT - 7T^{2} \)
11 \( 1 + 6.08T + 11T^{2} \)
17 \( 1 - 3.53iT - 17T^{2} \)
19 \( 1 + 7.05iT - 19T^{2} \)
23 \( 1 + 8.80T + 23T^{2} \)
29 \( 1 - 6.95iT - 29T^{2} \)
31 \( 1 - 7.51iT - 31T^{2} \)
37 \( 1 - 4.00T + 37T^{2} \)
41 \( 1 - 5.66iT - 41T^{2} \)
43 \( 1 + 11.6iT - 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 0.895iT - 53T^{2} \)
59 \( 1 + 0.110T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 + 4.96iT - 67T^{2} \)
71 \( 1 + 0.295T + 71T^{2} \)
73 \( 1 + 2.23T + 73T^{2} \)
79 \( 1 + 0.594iT - 79T^{2} \)
83 \( 1 - 2.65T + 83T^{2} \)
89 \( 1 - 4.54iT - 89T^{2} \)
97 \( 1 + 1.60T + 97T^{2} \)
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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

−9.212890707709592164852666810978, −8.611147452597665304875864399768, −7.81825389875834516842182061119, −6.61198920299582321685777241384, −5.80582104464872100053497306734, −5.19264665737663037319840040132, −4.63078913908137093752189658525, −3.32658772146367663371549604044, −2.56519719905935255688301882612, −1.74083488312454266141971409320, 0.04522035152394704423810448066, 2.13248299063798108832263081213, 3.12995160263332398416811426595, 4.02401754034995778783102115192, 4.64384832641463347458799355744, 5.81623018361187488179157942036, 6.18621060359543935124528483346, 7.39153870535132752681056945977, 7.82856931249540266460035142751, 8.125396141418745494162221642560

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