Properties

Label 2-360-1.1-c5-0-19
Degree $2$
Conductor $360$
Sign $-1$
Analytic cond. $57.7381$
Root an. cond. $7.59856$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 25·5-s + 94.1·7-s − 143.·11-s + 421.·13-s − 1.98e3·17-s − 1.31e3·19-s + 4.02e3·23-s + 625·25-s + 6.41e3·29-s − 2.35e3·31-s − 2.35e3·35-s − 7.87e3·37-s − 1.50e4·41-s + 1.14e3·43-s + 2.15e4·47-s − 7.94e3·49-s − 9.56e3·53-s + 3.59e3·55-s − 4.27e4·59-s + 3.21e4·61-s − 1.05e4·65-s − 3.03e4·67-s − 3.60e4·71-s − 6.34e4·73-s − 1.35e4·77-s − 8.99e4·79-s − 3.82e4·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.726·7-s − 0.358·11-s + 0.691·13-s − 1.66·17-s − 0.837·19-s + 1.58·23-s + 0.200·25-s + 1.41·29-s − 0.439·31-s − 0.324·35-s − 0.945·37-s − 1.40·41-s + 0.0941·43-s + 1.42·47-s − 0.472·49-s − 0.467·53-s + 0.160·55-s − 1.59·59-s + 1.10·61-s − 0.309·65-s − 0.826·67-s − 0.847·71-s − 1.39·73-s − 0.260·77-s − 1.62·79-s − 0.608·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-1$
Analytic conductor: \(57.7381\)
Root analytic conductor: \(7.59856\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 360,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
5 \( 1 + 25T \)
good7 \( 1 - 94.1T + 1.68e4T^{2} \)
11 \( 1 + 143.T + 1.61e5T^{2} \)
13 \( 1 - 421.T + 3.71e5T^{2} \)
17 \( 1 + 1.98e3T + 1.41e6T^{2} \)
19 \( 1 + 1.31e3T + 2.47e6T^{2} \)
23 \( 1 - 4.02e3T + 6.43e6T^{2} \)
29 \( 1 - 6.41e3T + 2.05e7T^{2} \)
31 \( 1 + 2.35e3T + 2.86e7T^{2} \)
37 \( 1 + 7.87e3T + 6.93e7T^{2} \)
41 \( 1 + 1.50e4T + 1.15e8T^{2} \)
43 \( 1 - 1.14e3T + 1.47e8T^{2} \)
47 \( 1 - 2.15e4T + 2.29e8T^{2} \)
53 \( 1 + 9.56e3T + 4.18e8T^{2} \)
59 \( 1 + 4.27e4T + 7.14e8T^{2} \)
61 \( 1 - 3.21e4T + 8.44e8T^{2} \)
67 \( 1 + 3.03e4T + 1.35e9T^{2} \)
71 \( 1 + 3.60e4T + 1.80e9T^{2} \)
73 \( 1 + 6.34e4T + 2.07e9T^{2} \)
79 \( 1 + 8.99e4T + 3.07e9T^{2} \)
83 \( 1 + 3.82e4T + 3.93e9T^{2} \)
89 \( 1 + 5.74e3T + 5.58e9T^{2} \)
97 \( 1 - 1.78e5T + 8.58e9T^{2} \)
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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

−10.47615774225553211203635226439, −8.850245100209165565418640370701, −8.527973656946793710134034916434, −7.28293833581298032159706799164, −6.41490095362630303132544092176, −5.03066402310484624616369917382, −4.24183654296505662337599131711, −2.84823631782948368784717679707, −1.49536660072773773982616003823, 0, 1.49536660072773773982616003823, 2.84823631782948368784717679707, 4.24183654296505662337599131711, 5.03066402310484624616369917382, 6.41490095362630303132544092176, 7.28293833581298032159706799164, 8.527973656946793710134034916434, 8.850245100209165565418640370701, 10.47615774225553211203635226439

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