Properties

Label 2-1560-1.1-c3-0-61
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 − 14.8·7-s + 9·9-s − 49.1·11-s + 13·13-s + 15·15-s − 27.0·17-s + 13.6·19-s − 44.5·21-s + 172.·23-s + 25·25-s + 27·27-s + 199.·29-s − 5.65·31-s − 147.·33-s − 74.2·35-s + 356.·37-s + 39·39-s − 219.·41-s − 542.·43-s + 45·45-s − 342.·47-s − 122.·49-s − 81.2·51-s − 355.·53-s − 245.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 0.801·7-s + 0.333·9-s − 1.34·11-s + 0.277·13-s + 0.258·15-s − 0.386·17-s + 0.164·19-s − 0.462·21-s + 1.56·23-s + 0.200·25-s + 0.192·27-s + 1.27·29-s − 0.0327·31-s − 0.777·33-s − 0.358·35-s + 1.58·37-s + 0.160·39-s − 0.837·41-s − 1.92·43-s + 0.149·45-s − 1.06·47-s − 0.357·49-s − 0.222·51-s − 0.921·53-s − 0.601·55-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 \( 1 + 14.8T + 343T^{2} \)
11 \( 1 + 49.1T + 1.33e3T^{2} \)
17 \( 1 + 27.0T + 4.91e3T^{2} \)
19 \( 1 - 13.6T + 6.85e3T^{2} \)
23 \( 1 - 172.T + 1.21e4T^{2} \)
29 \( 1 - 199.T + 2.43e4T^{2} \)
31 \( 1 + 5.65T + 2.97e4T^{2} \)
37 \( 1 - 356.T + 5.06e4T^{2} \)
41 \( 1 + 219.T + 6.89e4T^{2} \)
43 \( 1 + 542.T + 7.95e4T^{2} \)
47 \( 1 + 342.T + 1.03e5T^{2} \)
53 \( 1 + 355.T + 1.48e5T^{2} \)
59 \( 1 + 134.T + 2.05e5T^{2} \)
61 \( 1 - 188.T + 2.26e5T^{2} \)
67 \( 1 + 309.T + 3.00e5T^{2} \)
71 \( 1 + 73.1T + 3.57e5T^{2} \)
73 \( 1 + 696.T + 3.89e5T^{2} \)
79 \( 1 + 877.T + 4.93e5T^{2} \)
83 \( 1 + 1.46e3T + 5.71e5T^{2} \)
89 \( 1 - 984.T + 7.04e5T^{2} \)
97 \( 1 + 606.T + 9.12e5T^{2} \)
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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

−8.631817966849149392824575614314, −8.028498245499090363385533054948, −6.99403097410927963047837807668, −6.37806477762911473147331114695, −5.30188548077868048681934551547, −4.53740138147714388866472792721, −3.11054341301676490799985330037, −2.78468347581808261602305180278, −1.42350135201902743962023027999, 0, 1.42350135201902743962023027999, 2.78468347581808261602305180278, 3.11054341301676490799985330037, 4.53740138147714388866472792721, 5.30188548077868048681934551547, 6.37806477762911473147331114695, 6.99403097410927963047837807668, 8.028498245499090363385533054948, 8.631817966849149392824575614314

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