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Properties

Label 2-2340-15.8-c1-0-7
Degree $2$
Conductor $2340$
Sign $-0.767 - 0.640i$
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  + (−0.686 + 2.12i)5-s + (0.778 − 0.778i)7-s + 5.13i·11-s + (−0.707 − 0.707i)13-s + (2.51 + 2.51i)17-s + 0.758i·19-s + (−0.142 + 0.142i)23-s + (−4.05 − 2.92i)25-s + 2.35·29-s − 2.70·31-s + (1.12 + 2.19i)35-s + (−3.12 + 3.12i)37-s − 5.42i·41-s + (−1.79 − 1.79i)43-s + (−2.72 − 2.72i)47-s + ⋯
L(s)  = 1  + (−0.306 + 0.951i)5-s + (0.294 − 0.294i)7-s + 1.54i·11-s + (−0.196 − 0.196i)13-s + (0.610 + 0.610i)17-s + 0.174i·19-s + (−0.0296 + 0.0296i)23-s + (−0.811 − 0.584i)25-s + 0.438·29-s − 0.484·31-s + (0.189 + 0.370i)35-s + (−0.513 + 0.513i)37-s − 0.846i·41-s + (−0.273 − 0.273i)43-s + (−0.397 − 0.397i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 - 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.767 - 0.640i$
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: \((2,\ 2340,\ (\ :1/2),\ -0.767 - 0.640i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.135005071\)
\(L(\frac12)\) \(\approx\) \(1.135005071\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.686 - 2.12i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 + (-0.778 + 0.778i)T - 7iT^{2} \)
11 \( 1 - 5.13iT - 11T^{2} \)
17 \( 1 + (-2.51 - 2.51i)T + 17iT^{2} \)
19 \( 1 - 0.758iT - 19T^{2} \)
23 \( 1 + (0.142 - 0.142i)T - 23iT^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 2.70T + 31T^{2} \)
37 \( 1 + (3.12 - 3.12i)T - 37iT^{2} \)
41 \( 1 + 5.42iT - 41T^{2} \)
43 \( 1 + (1.79 + 1.79i)T + 43iT^{2} \)
47 \( 1 + (2.72 + 2.72i)T + 47iT^{2} \)
53 \( 1 + (7.53 - 7.53i)T - 53iT^{2} \)
59 \( 1 - 4.73T + 59T^{2} \)
61 \( 1 - 5.36T + 61T^{2} \)
67 \( 1 + (3.20 - 3.20i)T - 67iT^{2} \)
71 \( 1 - 5.57iT - 71T^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 - 15.2iT - 79T^{2} \)
83 \( 1 + (-8.68 + 8.68i)T - 83iT^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 + (2.00 - 2.00i)T - 97iT^{2} \)
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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

−9.435347131430245421081836459281, −8.349855977306479506241715297042, −7.58949306900821359745905363954, −7.12573018675138870626748097178, −6.33156624397029945017640348045, −5.31920516561995344734142695595, −4.39334891443888667475557120774, −3.62926560086096311449863343516, −2.57813160504481681051732727734, −1.56682061523228085031936099953, 0.38933475465444768005715158561, 1.53637973242715307827272970494, 2.92249257075655250521453430735, 3.77514141719317225518776870204, 4.84318164074413531247519123523, 5.42169993248896120661106479834, 6.22125739377480277169931997585, 7.27154274834328594433268484090, 8.170645593881762276282464291610, 8.560678244953081028793226271763

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