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Properties

Label 2-2340-15.8-c1-0-8
Degree $2$
Conductor $2340$
Sign $0.119 - 0.992i$
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  + (−2.15 + 0.583i)5-s + (−1.95 + 1.95i)7-s − 5.98i·11-s + (−0.707 − 0.707i)13-s + (3.87 + 3.87i)17-s + 0.340i·19-s + (5.93 − 5.93i)23-s + (4.31 − 2.51i)25-s − 1.93·29-s − 5.01·31-s + (3.08 − 5.36i)35-s + (−5.36 + 5.36i)37-s + 8.26i·41-s + (−1.30 − 1.30i)43-s + (9.14 + 9.14i)47-s + ⋯
L(s)  = 1  + (−0.965 + 0.260i)5-s + (−0.739 + 0.739i)7-s − 1.80i·11-s + (−0.196 − 0.196i)13-s + (0.940 + 0.940i)17-s + 0.0781i·19-s + (1.23 − 1.23i)23-s + (0.863 − 0.503i)25-s − 0.358·29-s − 0.900·31-s + (0.520 − 0.906i)35-s + (−0.882 + 0.882i)37-s + 1.29i·41-s + (−0.198 − 0.198i)43-s + (1.33 + 1.33i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.119 - 0.992i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.119 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9118870549\)
\(L(\frac12)\) \(\approx\) \(0.9118870549\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.583i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (1.95 - 1.95i)T - 7iT^{2} \)
11 \( 1 + 5.98iT - 11T^{2} \)
17 \( 1 + (-3.87 - 3.87i)T + 17iT^{2} \)
19 \( 1 - 0.340iT - 19T^{2} \)
23 \( 1 + (-5.93 + 5.93i)T - 23iT^{2} \)
29 \( 1 + 1.93T + 29T^{2} \)
31 \( 1 + 5.01T + 31T^{2} \)
37 \( 1 + (5.36 - 5.36i)T - 37iT^{2} \)
41 \( 1 - 8.26iT - 41T^{2} \)
43 \( 1 + (1.30 + 1.30i)T + 43iT^{2} \)
47 \( 1 + (-9.14 - 9.14i)T + 47iT^{2} \)
53 \( 1 + (-0.0960 + 0.0960i)T - 53iT^{2} \)
59 \( 1 + 0.114T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 + (5.13 - 5.13i)T - 67iT^{2} \)
71 \( 1 - 15.8iT - 71T^{2} \)
73 \( 1 + (-5.50 - 5.50i)T + 73iT^{2} \)
79 \( 1 - 6.60iT - 79T^{2} \)
83 \( 1 + (-0.738 + 0.738i)T - 83iT^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-5.79 + 5.79i)T - 97iT^{2} \)
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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

−8.870597823110625599642679855695, −8.512184225632180621159861488274, −7.74839150631862394840479124818, −6.78796333921538274590411221774, −6.04354472568896022201984707383, −5.39709133832732671477802756179, −4.18395248617841479366929858553, −3.21226810611970855875315139663, −2.87442770865019921246930321460, −0.979513080895891374244462070064, 0.39102814891464265859488220398, 1.81548995757013605203077596216, 3.24289932655424156106062368680, 3.86078914226152718892226643052, 4.79811351661231632781943334093, 5.44554080490869681154185507471, 6.98021778822419914619292752685, 7.23644500905909977717424005370, 7.68156195810818691713395947926, 9.113306325984695634098705319872

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