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Properties

Label 2-2340-39.11-c1-0-9
Degree $2$
Conductor $2340$
Sign $0.945 - 0.325i$
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 + (0.891 + 3.32i)7-s + (0.508 − 1.89i)11-s + (−2.17 − 2.87i)13-s + (−2.85 + 4.93i)17-s + (5.92 − 1.58i)19-s + (−2.03 − 3.52i)23-s − 1.00i·25-s + (3.49 − 2.01i)29-s + (6.96 + 6.96i)31-s + (2.98 + 1.72i)35-s + (6.75 + 1.80i)37-s + (8.34 + 2.23i)41-s + (1.37 + 0.793i)43-s + (−4.85 − 4.85i)47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + (0.336 + 1.25i)7-s + (0.153 − 0.572i)11-s + (−0.602 − 0.798i)13-s + (−0.691 + 1.19i)17-s + (1.35 − 0.363i)19-s + (−0.423 − 0.734i)23-s − 0.200i·25-s + (0.648 − 0.374i)29-s + (1.25 + 1.25i)31-s + (0.504 + 0.291i)35-s + (1.11 + 0.297i)37-s + (1.30 + 0.349i)41-s + (0.209 + 0.120i)43-s + (−0.707 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.945 - 0.325i$
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: \((2,\ 2340,\ (\ :1/2),\ 0.945 - 0.325i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.984617387\)
\(L(\frac12)\) \(\approx\) \(1.984617387\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 2.87i)T \)
good7 \( 1 + (-0.891 - 3.32i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.508 + 1.89i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.85 - 4.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.92 + 1.58i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.03 + 3.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.49 + 2.01i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.96 - 6.96i)T + 31iT^{2} \)
37 \( 1 + (-6.75 - 1.80i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-8.34 - 2.23i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.37 - 0.793i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.85 + 4.85i)T + 47iT^{2} \)
53 \( 1 + 7.98iT - 53T^{2} \)
59 \( 1 + (-6.45 + 1.72i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.91 - 8.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.21 - 8.26i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.02 - 11.2i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.23 - 8.23i)T - 73iT^{2} \)
79 \( 1 + 7.09T + 79T^{2} \)
83 \( 1 + (-10.5 + 10.5i)T - 83iT^{2} \)
89 \( 1 + (0.499 - 1.86i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.729 + 0.195i)T + (84.0 - 48.5i)T^{2} \)
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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

−8.704482181521254669986831566208, −8.555524739006166697317083405859, −7.66575146758649449553904792197, −6.49645482949149582658498565239, −5.86285473721399722686936235749, −5.16943494977889371195695887240, −4.36053314525628501745545908706, −3.00879057124174871415408350249, −2.34888217435632862354218908144, −1.01537137757480683907752129871, 0.866449044743518513524384297884, 2.08391300418068948217810422471, 3.12211455251332456116509554575, 4.34145745324567787231937857588, 4.69313385524596934031207221310, 5.90393900811468486690769066387, 6.77745622368719607991836593618, 7.49580387648783228708470129470, 7.80149383350284602353961159757, 9.392500798200399193032737590427

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