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Properties

Label 2-360-360.317-c1-0-50
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  + (−1 − i)2-s + (1.5 − 0.866i)3-s + 2i·4-s + (1.23 − 1.86i)5-s + (−2.36 − 0.633i)6-s + (2.86 − 0.767i)7-s + (2 − 2i)8-s + (1.5 − 2.59i)9-s + (−3.09 + 0.633i)10-s + (−1 + 1.73i)11-s + (1.73 + 3i)12-s + (−1.73 + 6.46i)13-s + (−3.63 − 2.09i)14-s + (0.232 − 3.86i)15-s − 4·16-s + (3 − 3i)17-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.866 − 0.499i)3-s + i·4-s + (0.550 − 0.834i)5-s + (−0.965 − 0.258i)6-s + (1.08 − 0.290i)7-s + (0.707 − 0.707i)8-s + (0.5 − 0.866i)9-s + (−0.979 + 0.200i)10-s + (−0.301 + 0.522i)11-s + (0.499 + 0.866i)12-s + (−0.480 + 1.79i)13-s + (−0.971 − 0.560i)14-s + (0.0599 − 0.998i)15-s − 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: $\chi_{360} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.0572 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07464 - 1.01478i\)
\(L(\frac12)\) \(\approx\) \(1.07464 - 1.01478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
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

−11.45655498784434833348179674028, −9.831742242276511173207760772313, −9.506244370844690522636513047723, −8.528102709195941318029128024516, −7.72301093173146191366578498356, −6.93932457750440931013950468415, −4.99143709089572192984765956535, −3.94133861164625572821403634187, −2.20072315159733612027983961372, −1.45941311144212231409615495502, 1.92776377028690140961719067085, 3.25813430030778326209470169939, 5.12578769731310175283882111040, 5.74910235235652587430443592300, 7.33540747503466395117742305568, 7.973233732471883398177020575854, 8.709015303329700392145288421036, 9.828464065665943147846062562769, 10.54162353936232752201001601622, 11.01948376294320741071976744157

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