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Properties

Label 2-2340-65.8-c1-0-6
Degree $2$
Conductor $2340$
Sign $0.180 - 0.983i$
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  + (−2.16 + 0.575i)5-s − 2.48i·7-s + (0.160 − 0.160i)11-s + (1.42 + 3.31i)13-s + (−1.58 − 1.58i)17-s + (−2.91 + 2.91i)19-s + (0.839 − 0.839i)23-s + (4.33 − 2.48i)25-s − 2.80i·29-s + (−3.31 − 3.31i)31-s + (1.43 + 5.37i)35-s + 5.33i·37-s + (6.07 + 6.07i)41-s + (−7.48 + 7.48i)43-s − 3.99i·47-s + ⋯
L(s)  = 1  + (−0.966 + 0.257i)5-s − 0.940i·7-s + (0.0484 − 0.0484i)11-s + (0.394 + 0.918i)13-s + (−0.384 − 0.384i)17-s + (−0.668 + 0.668i)19-s + (0.175 − 0.175i)23-s + (0.867 − 0.497i)25-s − 0.521i·29-s + (−0.594 − 0.594i)31-s + (0.242 + 0.908i)35-s + 0.877i·37-s + (0.948 + 0.948i)41-s + (−1.14 + 1.14i)43-s − 0.582i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.180 - 0.983i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9597278388\)
\(L(\frac12)\) \(\approx\) \(0.9597278388\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.16 - 0.575i)T \)
13 \( 1 + (-1.42 - 3.31i)T \)
good7 \( 1 + 2.48iT - 7T^{2} \)
11 \( 1 + (-0.160 + 0.160i)T - 11iT^{2} \)
17 \( 1 + (1.58 + 1.58i)T + 17iT^{2} \)
19 \( 1 + (2.91 - 2.91i)T - 19iT^{2} \)
23 \( 1 + (-0.839 + 0.839i)T - 23iT^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 + (3.31 + 3.31i)T + 31iT^{2} \)
37 \( 1 - 5.33iT - 37T^{2} \)
41 \( 1 + (-6.07 - 6.07i)T + 41iT^{2} \)
43 \( 1 + (7.48 - 7.48i)T - 43iT^{2} \)
47 \( 1 + 3.99iT - 47T^{2} \)
53 \( 1 + (-0.0154 - 0.0154i)T + 53iT^{2} \)
59 \( 1 + (-2.91 - 2.91i)T + 59iT^{2} \)
61 \( 1 + 7.60T + 61T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (-11.3 - 11.3i)T + 71iT^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 15.6iT - 79T^{2} \)
83 \( 1 - 0.681iT - 83T^{2} \)
89 \( 1 + (-8.11 - 8.11i)T + 89iT^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−9.130516527222555362734838895052, −8.190216727398846327191666270378, −7.73213593608566039087222049817, −6.75497933038893370344295859589, −6.39816018070566217918658101054, −4.99294502088959546050738225756, −4.13263789788487497525685261605, −3.71715422133851576447885579789, −2.46469514554471908668755483021, −1.05966480293240283895597005979, 0.38370709402166111179329039102, 1.95136773366084902915264933288, 3.09126888073315197464183623347, 3.87774433696388892685188129693, 4.89937582088459672393924643807, 5.57332577683523537781312158658, 6.51715208814264398208045957309, 7.37725819538151374446775153170, 8.140861848821469862484493525449, 8.848565051058825396891956148360

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