Properties

Label 2-1560-1.1-c3-0-52
Degree $2$
Conductor $1560$
Sign $-1$
Analytic cond. $92.0429$
Root an. cond. $9.59390$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5·5-s + 23.2·7-s + 9·9-s + 49.6·11-s − 13·13-s + 15·15-s − 17.0·17-s − 129.·19-s − 69.6·21-s − 149.·23-s + 25·25-s − 27·27-s + 11.3·29-s − 75.4·31-s − 148.·33-s − 116.·35-s + 84.5·37-s + 39·39-s + 136.·41-s − 240.·43-s − 45·45-s + 5.99·47-s + 196.·49-s + 51.1·51-s + 25.9·53-s − 248.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1.25·7-s + 0.333·9-s + 1.36·11-s − 0.277·13-s + 0.258·15-s − 0.243·17-s − 1.55·19-s − 0.723·21-s − 1.35·23-s + 0.200·25-s − 0.192·27-s + 0.0725·29-s − 0.437·31-s − 0.785·33-s − 0.560·35-s + 0.375·37-s + 0.160·39-s + 0.518·41-s − 0.854·43-s − 0.149·45-s + 0.0185·47-s + 0.571·49-s + 0.140·51-s + 0.0673·53-s − 0.608·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-1$
Analytic conductor: \(92.0429\)
Root analytic conductor: \(9.59390\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1560,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 3T \)
5 \( 1 + 5T \)
13 \( 1 + 13T \)
good7 \( 1 - 23.2T + 343T^{2} \)
11 \( 1 - 49.6T + 1.33e3T^{2} \)
17 \( 1 + 17.0T + 4.91e3T^{2} \)
19 \( 1 + 129.T + 6.85e3T^{2} \)
23 \( 1 + 149.T + 1.21e4T^{2} \)
29 \( 1 - 11.3T + 2.43e4T^{2} \)
31 \( 1 + 75.4T + 2.97e4T^{2} \)
37 \( 1 - 84.5T + 5.06e4T^{2} \)
41 \( 1 - 136.T + 6.89e4T^{2} \)
43 \( 1 + 240.T + 7.95e4T^{2} \)
47 \( 1 - 5.99T + 1.03e5T^{2} \)
53 \( 1 - 25.9T + 1.48e5T^{2} \)
59 \( 1 - 386.T + 2.05e5T^{2} \)
61 \( 1 - 476.T + 2.26e5T^{2} \)
67 \( 1 - 298.T + 3.00e5T^{2} \)
71 \( 1 + 663.T + 3.57e5T^{2} \)
73 \( 1 + 99.1T + 3.89e5T^{2} \)
79 \( 1 - 480.T + 4.93e5T^{2} \)
83 \( 1 - 397.T + 5.71e5T^{2} \)
89 \( 1 + 1.31e3T + 7.04e5T^{2} \)
97 \( 1 + 194.T + 9.12e5T^{2} \)
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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

−8.518676119977125003815950192380, −7.988990592694534793606883726736, −6.96172647867344658739160258685, −6.30247643373145379031681826543, −5.31979655437724279948806287512, −4.32519990976052328912840829037, −3.95253382604192390448869248035, −2.21322549307073211445083538119, −1.32097446149692544758585933941, 0, 1.32097446149692544758585933941, 2.21322549307073211445083538119, 3.95253382604192390448869248035, 4.32519990976052328912840829037, 5.31979655437724279948806287512, 6.30247643373145379031681826543, 6.96172647867344658739160258685, 7.988990592694534793606883726736, 8.518676119977125003815950192380

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