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Properties

Label 2-2340-12.11-c1-0-15
Degree $2$
Conductor $2340$
Sign $-0.782 + 0.622i$
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  + (0.849 + 1.13i)2-s + (−0.558 + 1.92i)4-s + i·5-s + 2.23i·7-s + (−2.64 + 0.998i)8-s + (−1.13 + 0.849i)10-s − 0.882·11-s + 13-s + (−2.52 + 1.89i)14-s + (−3.37 − 2.14i)16-s + 5.34i·17-s − 2.54i·19-s + (−1.92 − 0.558i)20-s + (−0.749 − 0.997i)22-s − 2.87·23-s + ⋯
L(s)  = 1  + (0.600 + 0.799i)2-s + (−0.279 + 0.960i)4-s + 0.447i·5-s + 0.845i·7-s + (−0.935 + 0.353i)8-s + (−0.357 + 0.268i)10-s − 0.266·11-s + 0.277·13-s + (−0.676 + 0.507i)14-s + (−0.844 − 0.536i)16-s + 1.29i·17-s − 0.584i·19-s + (−0.429 − 0.124i)20-s + (−0.159 − 0.212i)22-s − 0.600·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.782 + 0.622i$
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: \((2,\ 2340,\ (\ :1/2),\ -0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.389186180\)
\(L(\frac12)\) \(\approx\) \(1.389186180\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.849 - 1.13i)T \)
3 \( 1 \)
5 \( 1 - iT \)
13 \( 1 - T \)
good7 \( 1 - 2.23iT - 7T^{2} \)
11 \( 1 + 0.882T + 11T^{2} \)
17 \( 1 - 5.34iT - 17T^{2} \)
19 \( 1 + 2.54iT - 19T^{2} \)
23 \( 1 + 2.87T + 23T^{2} \)
29 \( 1 + 3.71iT - 29T^{2} \)
31 \( 1 - 9.00iT - 31T^{2} \)
37 \( 1 + 7.25T + 37T^{2} \)
41 \( 1 - 9.48iT - 41T^{2} \)
43 \( 1 + 5.96iT - 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 + 7.55iT - 53T^{2} \)
59 \( 1 + 11.0T + 59T^{2} \)
61 \( 1 - 6.51T + 61T^{2} \)
67 \( 1 + 8.70iT - 67T^{2} \)
71 \( 1 - 5.02T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 - 2.63iT - 79T^{2} \)
83 \( 1 + 15.6T + 83T^{2} \)
89 \( 1 + 0.165iT - 89T^{2} \)
97 \( 1 - 4.41T + 97T^{2} \)
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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

−9.185535810449652090214725082722, −8.440643351900425355780944938722, −7.989007736374943171356919662937, −6.90881214002703584725948240247, −6.35766363165474268885356748894, −5.60763185078296438728213285783, −4.87164443365818019870679463967, −3.82212158050236460600404539121, −3.04586097103781544055516060213, −1.99527392072235581734682754313, 0.37127463379220996797120740367, 1.50900362959119864238172812269, 2.64634448955043605805602820226, 3.69925151843705219308004489646, 4.33665095904376332951658607552, 5.21564089589193296836594413940, 5.90731776226857142888829455382, 6.93719218533819206004434997176, 7.71689555574580461560035544485, 8.714756548246326527698582806659

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