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Properties

Label 2-2340-39.11-c1-0-10
Degree $2$
Conductor $2340$
Sign $0.710 - 0.703i$
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  + (−0.707 + 0.707i)5-s + (1.29 + 4.83i)7-s + (1.26 − 4.72i)11-s + (3.20 + 1.65i)13-s + (1.47 − 2.55i)17-s + (3.97 − 1.06i)19-s + (0.451 + 0.782i)23-s − 1.00i·25-s + (4.66 − 2.69i)29-s + (−4.05 − 4.05i)31-s + (−4.33 − 2.50i)35-s + (−4.50 − 1.20i)37-s + (6.53 + 1.74i)41-s + (5.10 + 2.94i)43-s + (3.65 + 3.65i)47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + (0.489 + 1.82i)7-s + (0.381 − 1.42i)11-s + (0.888 + 0.458i)13-s + (0.357 − 0.620i)17-s + (0.912 − 0.244i)19-s + (0.0941 + 0.163i)23-s − 0.200i·25-s + (0.865 − 0.499i)29-s + (−0.727 − 0.727i)31-s + (−0.732 − 0.422i)35-s + (−0.740 − 0.198i)37-s + (1.01 + 0.273i)41-s + (0.778 + 0.449i)43-s + (0.532 + 0.532i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 - 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.710 - 0.703i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.710 - 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.001198961\)
\(L(\frac12)\) \(\approx\) \(2.001198961\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + (-3.20 - 1.65i)T \)
good7 \( 1 + (-1.29 - 4.83i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.26 + 4.72i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.47 + 2.55i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.97 + 1.06i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.451 - 0.782i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.66 + 2.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.05 + 4.05i)T + 31iT^{2} \)
37 \( 1 + (4.50 + 1.20i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-6.53 - 1.74i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.10 - 2.94i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.65 - 3.65i)T + 47iT^{2} \)
53 \( 1 - 9.56iT - 53T^{2} \)
59 \( 1 + (-8.99 + 2.40i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.81 + 4.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.99 - 7.44i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.55 + 9.55i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (3.73 - 3.73i)T - 73iT^{2} \)
79 \( 1 - 9.42T + 79T^{2} \)
83 \( 1 + (10.7 - 10.7i)T - 83iT^{2} \)
89 \( 1 + (2.43 - 9.07i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.38 - 0.908i)T + (84.0 - 48.5i)T^{2} \)
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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

−9.091658728197785900245048744893, −8.373094774699900737977171176698, −7.75058839391184295262516528971, −6.61879833629045875329078281610, −5.80262050413709660061284920066, −5.43942136583767452444657684244, −4.19901759177592128392559284068, −3.16742845587308146909943834516, −2.48077446453219159254306763326, −1.08859202138920913822611031046, 0.896748451759690631305485718710, 1.68115722021004055437219189813, 3.41400594586416185468215055179, 4.04326444241855113464071810519, 4.73354304987379534347107375209, 5.65938474525341131724931430571, 6.98862505997137273409128827410, 7.19997583129971157907726307736, 8.057782401421424537010332290802, 8.765376609482382930322620034656

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