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Properties

Label 2-2340-12.11-c1-0-16
Degree $2$
Conductor $2340$
Sign $-0.684 + 0.729i$
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  + (0.142 + 1.40i)2-s + (−1.95 + 0.400i)4-s + i·5-s + 4.04i·7-s + (−0.842 − 2.70i)8-s + (−1.40 + 0.142i)10-s + 2.27·11-s − 13-s + (−5.69 + 0.576i)14-s + (3.67 − 1.56i)16-s + 3.42i·17-s + 0.567i·19-s + (−0.400 − 1.95i)20-s + (0.324 + 3.20i)22-s − 1.96·23-s + ⋯
L(s)  = 1  + (0.100 + 0.994i)2-s + (−0.979 + 0.200i)4-s + 0.447i·5-s + 1.53i·7-s + (−0.297 − 0.954i)8-s + (−0.444 + 0.0450i)10-s + 0.686·11-s − 0.277·13-s + (−1.52 + 0.154i)14-s + (0.919 − 0.392i)16-s + 0.830i·17-s + 0.130i·19-s + (−0.0895 − 0.438i)20-s + (0.0690 + 0.683i)22-s − 0.408·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.684 + 0.729i$
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: \((2,\ 2340,\ (\ :1/2),\ -0.684 + 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.110744069\)
\(L(\frac12)\) \(\approx\) \(1.110744069\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.142 - 1.40i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 + T \)
good7 \( 1 - 4.04iT - 7T^{2} \)
11 \( 1 - 2.27T + 11T^{2} \)
17 \( 1 - 3.42iT - 17T^{2} \)
19 \( 1 - 0.567iT - 19T^{2} \)
23 \( 1 + 1.96T + 23T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 - 8.13iT - 31T^{2} \)
37 \( 1 + 4.83T + 37T^{2} \)
41 \( 1 - 5.67iT - 41T^{2} \)
43 \( 1 + 9.24iT - 43T^{2} \)
47 \( 1 + 1.33T + 47T^{2} \)
53 \( 1 + 0.876iT - 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 2.03T + 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 - 4.49T + 71T^{2} \)
73 \( 1 + 14.2T + 73T^{2} \)
79 \( 1 + 6.76iT - 79T^{2} \)
83 \( 1 - 5.39T + 83T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 + 4.37T + 97T^{2} \)
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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

−9.142344666998440036920961150682, −8.667131742743878868930030243484, −8.009733012592047401013722242431, −6.95240483860254135865975677396, −6.42559637462081090094401238895, −5.63300176250319990061282777234, −5.02549159666278911755967111734, −3.88352944173056631450445694715, −3.02975824732293393935689973858, −1.73827656349216338593644302809, 0.39070374118525139456087811185, 1.31765481804187298223707639004, 2.52662762502391246120329074586, 3.75178942779969567666210405020, 4.21712374954971995807602288706, 5.00279836836006602893280696853, 6.03573089705181634844128705917, 7.10491721064599468706858938821, 7.80535233773966071054063671505, 8.679649067776355244563929675813

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