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Properties

Label 2-2340-39.5-c1-0-1
Degree $2$
Conductor $2340$
Sign $-0.959 - 0.282i$
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 + (−1.71 − 1.71i)7-s + (−2.24 + 2.24i)11-s + (2.45 − 2.63i)13-s − 5.14·17-s + 4.89·23-s + 1.00i·25-s + 3.07i·29-s + (−4.91 + 4.91i)31-s − 2.42i·35-s + (−5.37 − 5.37i)37-s + (−6.37 − 6.37i)41-s + 10.3i·43-s + (0.764 − 0.764i)47-s − 1.09i·49-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + (−0.649 − 0.649i)7-s + (−0.677 + 0.677i)11-s + (0.682 − 0.731i)13-s − 1.24·17-s + 1.02·23-s + 0.200i·25-s + 0.571i·29-s + (−0.883 + 0.883i)31-s − 0.410i·35-s + (−0.884 − 0.884i)37-s + (−0.995 − 0.995i)41-s + 1.57i·43-s + (0.111 − 0.111i)47-s − 0.156i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.959 - 0.282i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ -0.959 - 0.282i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2154767834\)
\(L(\frac12)\) \(\approx\) \(0.2154767834\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 + 2.63i)T \)
good7 \( 1 + (1.71 + 1.71i)T + 7iT^{2} \)
11 \( 1 + (2.24 - 2.24i)T - 11iT^{2} \)
17 \( 1 + 5.14T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 4.89T + 23T^{2} \)
29 \( 1 - 3.07iT - 29T^{2} \)
31 \( 1 + (4.91 - 4.91i)T - 31iT^{2} \)
37 \( 1 + (5.37 + 5.37i)T + 37iT^{2} \)
41 \( 1 + (6.37 + 6.37i)T + 41iT^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (-0.764 + 0.764i)T - 47iT^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + (8.98 - 8.98i)T - 59iT^{2} \)
61 \( 1 + 13.9T + 61T^{2} \)
67 \( 1 + (-5.43 + 5.43i)T - 67iT^{2} \)
71 \( 1 + (-9.56 - 9.56i)T + 71iT^{2} \)
73 \( 1 + (11.4 + 11.4i)T + 73iT^{2} \)
79 \( 1 + 5.57T + 79T^{2} \)
83 \( 1 + (1.13 + 1.13i)T + 83iT^{2} \)
89 \( 1 + (-2.36 + 2.36i)T - 89iT^{2} \)
97 \( 1 + (2.97 - 2.97i)T - 97iT^{2} \)
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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

−9.217639137342895128630500365164, −8.765819882840896292605073310380, −7.58372335748666907698208640966, −7.06471296380675924443453885207, −6.34369287510336824822215726345, −5.41305429897917397475368451114, −4.58139278321892867184018191859, −3.51662872408635391967489964250, −2.77161054544584852834528285909, −1.53027247604696081206216313398, 0.06898545807669371809415884370, 1.72286110434525696618118580907, 2.72741624660827136574963887968, 3.64996518677316607559981717703, 4.73714366403407226785728842664, 5.53281327878389553264353794457, 6.32607151245427721411413233048, 6.88271254216404140326801821794, 8.070335844795801935815830127897, 8.792205390970555252125046120171

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