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Properties

Label 2-1140-1140.599-c0-0-0
Degree $2$
Conductor $1140$
Sign $0.624 - 0.780i$
Analytic cond. $0.568934$
Root an. cond. $0.754277$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.499 + 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.939 + 0.342i)6-s + 0.999·8-s + (0.173 − 0.984i)9-s + (0.173 + 0.984i)10-s + (−0.173 − 0.984i)12-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (−0.0603 − 0.342i)17-s + (−0.939 + 0.342i)18-s + (−0.766 + 0.642i)19-s + (0.766 − 0.642i)20-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.766 + 0.642i)3-s + (−0.499 + 0.866i)4-s + (−0.939 − 0.342i)5-s + (0.939 + 0.342i)6-s + 0.999·8-s + (0.173 − 0.984i)9-s + (0.173 + 0.984i)10-s + (−0.173 − 0.984i)12-s + (0.939 − 0.342i)15-s + (−0.5 − 0.866i)16-s + (−0.0603 − 0.342i)17-s + (−0.939 + 0.342i)18-s + (−0.766 + 0.642i)19-s + (0.766 − 0.642i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1140\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.624 - 0.780i$
Analytic conductor: \(0.568934\)
Root analytic conductor: \(0.754277\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1140} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1140,\ (\ :0),\ 0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3725598614\)
\(L(\frac12)\) \(\approx\) \(0.3725598614\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.766 - 0.642i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.0603 + 0.342i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.592 - 1.62i)T + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (1.11 + 0.642i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T^{2} \)
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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

−10.38489202492094992405968923982, −9.236756071210595058282945998491, −8.903307278561699948840025974197, −7.74303259678923685061347811247, −7.05976512456902775433881981892, −5.65374773080616722227824345279, −4.70564720478593961797452939966, −3.94165157628262377410746774775, −3.13009943882289156099501035238, −1.28246070408174254406344782929, 0.49378954245293517401347192618, 2.26103916336126991944241322963, 4.09130818769012600444836110483, 4.90164545921876079949171740078, 5.94679782726489460824637562575, 6.77326538040177776509557482247, 7.21574026881180982814979064905, 8.188829198596613942711969475028, 8.677114544961608983825674126068, 9.938210430673898935515443013404

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