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Properties

Label 2-360-360.293-c1-0-10
Degree $2$
Conductor $360$
Sign $0.0572 - 0.998i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
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  + (0.366 + 1.36i)2-s + (−1.5 − 0.866i)3-s + (−1.73 + i)4-s + (−1.23 − 1.86i)5-s + (0.633 − 2.36i)6-s + (2.86 + 0.767i)7-s + (−2 − 1.99i)8-s + (1.5 + 2.59i)9-s + (2.09 − 2.36i)10-s + (1 + 1.73i)11-s + 3.46·12-s + (1.73 + 6.46i)13-s + 4.19i·14-s + (0.232 + 3.86i)15-s + (1.99 − 3.46i)16-s + (3 + 3i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 − 0.499i)3-s + (−0.866 + 0.5i)4-s + (−0.550 − 0.834i)5-s + (0.258 − 0.965i)6-s + (1.08 + 0.290i)7-s + (−0.707 − 0.707i)8-s + (0.5 + 0.866i)9-s + (0.663 − 0.748i)10-s + (0.301 + 0.522i)11-s + 0.999·12-s + (0.480 + 1.79i)13-s + 1.12i·14-s + (0.0599 + 0.998i)15-s + (0.499 − 0.866i)16-s + (0.727 + 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0572 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.0572 - 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.0572 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.759891 + 0.717558i\)
\(L(\frac12)\) \(\approx\) \(0.759891 + 0.717558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
good7 \( 1 + (-2.86 - 0.767i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - 6.46i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (-3 - 3i)T + 17iT^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + (0.866 + 3.23i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-6.23 + 3.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.09 - 1.90i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.46 - 6.46i)T + 37iT^{2} \)
41 \( 1 + (0.401 + 0.232i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.633 - 0.169i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.5 - 5.59i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-5.19 - 5.19i)T + 53iT^{2} \)
59 \( 1 + (10.5 + 6.09i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.30 + 1.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.96 - 1.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + (-0.196 - 0.196i)T + 73iT^{2} \)
79 \( 1 + (-7.09 + 4.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.66 - 9.96i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + (6.83 + 1.83i)T + (84.0 + 48.5i)T^{2} \)
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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

−12.05973512327466429318938714306, −11.06710674904881158640674778801, −9.573107064836488631114490803324, −8.459482276993593418554073566031, −7.913883925498489434741448411608, −6.78657717347621211806428700104, −5.93941032462266938949121957462, −4.66381446021428820949269214582, −4.35079377124842871626522163057, −1.47594846873974213564243768243, 0.874667160316508088734724658460, 3.06568392579356641276332740867, 4.00736884418332958760259752986, 5.16284364332135599421770305333, 5.98720574934315442036585138748, 7.52668133071366161070909193408, 8.518970924150077383733327624850, 9.922680203247595101515234818809, 10.61257021843154690252651615683, 11.18214449974130918213782817617

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