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Properties

Label 2-2340-13.10-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.0183 - 0.999i$
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  − i·5-s + (3.75 + 2.16i)7-s + (−1.5 + 0.866i)11-s + (−3.11 − 1.81i)13-s + (−3.75 + 6.49i)17-s + (4.65 + 2.68i)19-s + (0.580 + 1.00i)23-s − 25-s + (−1.01 − 1.75i)29-s + 7.86i·31-s + (2.16 − 3.75i)35-s + (−8.25 + 4.76i)37-s + (−6.69 + 3.86i)41-s + (2.09 − 3.62i)43-s − 3.46i·47-s + ⋯
L(s)  = 1  − 0.447i·5-s + (1.41 + 0.818i)7-s + (−0.452 + 0.261i)11-s + (−0.863 − 0.504i)13-s + (−0.909 + 1.57i)17-s + (1.06 + 0.616i)19-s + (0.121 + 0.209i)23-s − 0.200·25-s + (−0.187 − 0.325i)29-s + 1.41i·31-s + (0.366 − 0.634i)35-s + (−1.35 + 0.783i)37-s + (−1.04 + 0.603i)41-s + (0.319 − 0.552i)43-s − 0.505i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0183 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.0183 - 0.999i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.0183 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.545112233\)
\(L(\frac12)\) \(\approx\) \(1.545112233\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + iT \)
13 \( 1 + (3.11 + 1.81i)T \)
good7 \( 1 + (-3.75 - 2.16i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 0.866i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.75 - 6.49i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.65 - 2.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.580 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.01 + 1.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.86iT - 31T^{2} \)
37 \( 1 + (8.25 - 4.76i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.69 - 3.86i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.09 + 3.62i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 + (5.49 + 3.17i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.85 - 3.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.55 - 2.63i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.8 - 6.25i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 5.23iT - 73T^{2} \)
79 \( 1 + 8.16T + 79T^{2} \)
83 \( 1 - 0.456iT - 83T^{2} \)
89 \( 1 + (-11.4 + 6.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.43 + 1.40i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.877136064611095856661589467395, −8.463647934406847022787943140513, −7.82001338035085302520456196218, −6.99789645707214973702020589549, −5.79065427587769642796769635508, −5.18225639567922743596296999538, −4.64284015304572472447476489750, −3.45032400570871732344200245496, −2.19790727254692922148629821142, −1.47776771459947422870309231032, 0.51371757724103574672679695746, 1.96428241252385592707180600001, 2.83049186614492714499743631491, 4.06747792630849771328685999325, 4.88225686992551987572719142252, 5.36848917740848284651872950764, 6.77817712810029895081391452276, 7.34222901139307698507276665471, 7.76864298218381497566519480135, 8.859423102731440457911656858977

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