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Properties

Label 2-1560-13.3-c1-0-18
Degree $2$
Conductor $1560$
Sign $0.923 + 0.384i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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  + (0.5 + 0.866i)3-s + 5-s + (−0.219 + 0.379i)7-s + (−0.499 + 0.866i)9-s + (−2.28 − 3.95i)11-s + (3.34 − 1.35i)13-s + (0.5 + 0.866i)15-s + (1 − 1.73i)17-s + (2.56 − 4.43i)19-s − 0.438·21-s + (−2.28 − 3.95i)23-s + 25-s − 0.999·27-s + (1.84 + 3.19i)29-s + 4.12·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + 0.447·5-s + (−0.0828 + 0.143i)7-s + (−0.166 + 0.288i)9-s + (−0.687 − 1.19i)11-s + (0.926 − 0.375i)13-s + (0.129 + 0.223i)15-s + (0.242 − 0.420i)17-s + (0.587 − 1.01i)19-s − 0.0956·21-s + (−0.475 − 0.823i)23-s + 0.200·25-s − 0.192·27-s + (0.342 + 0.592i)29-s + 0.740·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.923 + 0.384i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.923 + 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.974423568\)
\(L(\frac12)\) \(\approx\) \(1.974423568\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
13 \( 1 + (-3.34 + 1.35i)T \)
good7 \( 1 + (0.219 - 0.379i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.28 + 3.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.56 + 4.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.28 + 3.95i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.84 - 3.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.12T + 31T^{2} \)
37 \( 1 + (2.28 + 3.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.06 + 5.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.56T + 47T^{2} \)
53 \( 1 - 1.12T + 53T^{2} \)
59 \( 1 + (-1.84 + 3.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.34 - 9.25i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.21 - 2.11i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 4.43T + 73T^{2} \)
79 \( 1 + 0.753T + 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.21 - 7.30i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.230606470515877938801277334345, −8.716109920000913276151854371931, −8.002083887928085280189774051211, −6.96416082847718841265155289721, −5.89867677806275475236165706360, −5.40477755729558784142267734614, −4.32267322237057787874704424572, −3.19451522709642097795306839542, −2.56146943700815230721800551845, −0.821826898942895902373244117234, 1.33440431811206201798467641212, 2.23494200636166887441195859057, 3.42369503226892033697298010121, 4.40106488290895596183398120865, 5.55003029427577527971373316668, 6.23645939373602552606492170878, 7.16547499544664678738653354909, 7.87727346599992886548882729735, 8.563746893971433034134819685007, 9.662372757456259049738045808430

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