Properties

Label 2-1560-5.4-c1-0-1
Degree $2$
Conductor $1560$
Sign $0.158 - 0.987i$
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  i·3-s + (−2.20 − 0.353i)5-s − 1.65i·7-s − 9-s − 2.94·11-s + i·13-s + (−0.353 + 2.20i)15-s + 1.46i·17-s − 1.65·21-s − 0.532i·23-s + (4.74 + 1.56i)25-s + i·27-s − 5.70·29-s + 2.94i·33-s + (−0.585 + 3.65i)35-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.987 − 0.158i)5-s − 0.625i·7-s − 0.333·9-s − 0.888·11-s + 0.277i·13-s + (−0.0913 + 0.570i)15-s + 0.355i·17-s − 0.361·21-s − 0.111i·23-s + (0.949 + 0.312i)25-s + 0.192i·27-s − 1.06·29-s + 0.513i·33-s + (−0.0989 + 0.617i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.158 - 0.987i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.158 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5091734911\)
\(L(\frac12)\) \(\approx\) \(0.5091734911\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 + (2.20 + 0.353i)T \)
13 \( 1 - iT \)
good7 \( 1 + 1.65iT - 7T^{2} \)
11 \( 1 + 2.94T + 11T^{2} \)
17 \( 1 - 1.46iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 0.532iT - 23T^{2} \)
29 \( 1 + 5.70T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 8.77iT - 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 - 1.70iT - 43T^{2} \)
47 \( 1 - 2.70iT - 47T^{2} \)
53 \( 1 - 8.77iT - 53T^{2} \)
59 \( 1 - 3.83T + 59T^{2} \)
61 \( 1 + 0.241T + 61T^{2} \)
67 \( 1 - 2.58iT - 67T^{2} \)
71 \( 1 + 2.55T + 71T^{2} \)
73 \( 1 + 0.188iT - 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 - 7.91iT - 83T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 - 16.8iT - 97T^{2} \)
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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

−9.570464018123416972949316395033, −8.578530172099644741760716897644, −7.911769787457811455367846601904, −7.35893864226951915312736743114, −6.58110484223890047343384518172, −5.50052427995353639584291080788, −4.53428465509764037398173187037, −3.68103080528412529963105124144, −2.61306449419521224583889078974, −1.16960671105137827370954718128, 0.21858177390418728187093117875, 2.31420867604760437730410646515, 3.27967583110878382253294619736, 4.12464807928642992270040862500, 5.14494671945012328141402063142, 5.74555070725823687404137994974, 7.01426009182304390297347416298, 7.73148144955919080357355233839, 8.479101107061021221173788106718, 9.199128288021881850708734928207

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