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Properties

Label 2-2340-39.32-c1-0-8
Degree $2$
Conductor $2340$
Sign $0.990 - 0.137i$
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 + (−0.420 + 1.57i)7-s + (0.0781 + 0.291i)11-s + (0.635 − 3.54i)13-s + (1.26 + 2.19i)17-s + (−4.60 − 1.23i)19-s + (−2.25 + 3.90i)23-s + 1.00i·25-s + (8.42 + 4.86i)29-s + (6.01 − 6.01i)31-s + (1.40 − 0.812i)35-s + (9.00 − 2.41i)37-s + (−3.76 + 1.00i)41-s + (−2.10 + 1.21i)43-s + (4.45 − 4.45i)47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + (−0.159 + 0.593i)7-s + (0.0235 + 0.0879i)11-s + (0.176 − 0.984i)13-s + (0.307 + 0.532i)17-s + (−1.05 − 0.282i)19-s + (−0.470 + 0.814i)23-s + 0.200i·25-s + (1.56 + 0.903i)29-s + (1.07 − 1.07i)31-s + (0.237 − 0.137i)35-s + (1.47 − 0.396i)37-s + (−0.587 + 0.157i)41-s + (−0.321 + 0.185i)43-s + (0.649 − 0.649i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.990 - 0.137i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.990 - 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.566702150\)
\(L(\frac12)\) \(\approx\) \(1.566702150\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.635 + 3.54i)T \)
good7 \( 1 + (0.420 - 1.57i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-0.0781 - 0.291i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.26 - 2.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.60 + 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.25 - 3.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.42 - 4.86i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.01 + 6.01i)T - 31iT^{2} \)
37 \( 1 + (-9.00 + 2.41i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.76 - 1.00i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.10 - 1.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.45 + 4.45i)T - 47iT^{2} \)
53 \( 1 - 5.50iT - 53T^{2} \)
59 \( 1 + (6.04 + 1.61i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.78 + 3.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.318 + 1.18i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.47 + 5.51i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.50 - 1.50i)T + 73iT^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + (-5.60 - 5.60i)T + 83iT^{2} \)
89 \( 1 + (-3.34 - 12.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-12.5 - 3.35i)T + (84.0 + 48.5i)T^{2} \)
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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

−8.930643610492541568888549858673, −8.169543818761940175460692351265, −7.74173821707405467277763830516, −6.50789011999235451822465321383, −5.95059296386277072561171788200, −5.04024638328106553603102769012, −4.19469440724064938959937273402, −3.20755958913629534668462301812, −2.25881373729574756653131707982, −0.850742392929427064729660150180, 0.76791403294037542200134496548, 2.21193651109795296424982849236, 3.20250637354068858741698367313, 4.27276496026698301808276982297, 4.68249607862312957069194679683, 6.18480603528796760085687591953, 6.54332995853545976048579481709, 7.40350040734429018673791514731, 8.258314914605254052647947320289, 8.818140098330928656573018850335

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