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Properties

Label 2-360-360.173-c1-0-51
Degree $2$
Conductor $360$
Sign $-0.395 + 0.918i$
Analytic cond. $2.87461$
Root an. cond. $1.69546$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + i)2-s + (0.866 + 1.5i)3-s − 2i·4-s + (−2.23 + 0.133i)5-s + (−2.36 − 0.633i)6-s + (−0.767 − 2.86i)7-s + (2 + 2i)8-s + (−1.5 + 2.59i)9-s + (2.09 − 2.36i)10-s + (−1 + 1.73i)11-s + (3 − 1.73i)12-s + (−6.46 − 1.73i)13-s + (3.63 + 2.09i)14-s + (−2.13 − 3.23i)15-s − 4·16-s + (−3 − 3i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.499 + 0.866i)3-s − i·4-s + (−0.998 + 0.0599i)5-s + (−0.965 − 0.258i)6-s + (−0.290 − 1.08i)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 + (0.866 − 0.499i)12-s + (−1.79 − 0.480i)13-s + (0.971 + 0.560i)14-s + (−0.550 − 0.834i)15-s − 16-s + (−0.727 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.395 + 0.918i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (2.23 - 0.133i)T \)
good7 \( 1 + (0.767 + 2.86i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (6.46 + 1.73i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + (3.23 + 0.866i)T + (19.9 + 11.5i)T^{2} \)
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 \( 1 + (0.169 + 0.633i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (5.59 - 1.5i)T + (40.7 - 23.5i)T^{2} \)
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 \( 1 + (-1.33 + 4.96i)T + (-58.0 - 33.5i)T^{2} \)
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 \( 1 + (9.96 - 2.66i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 + (-1.83 - 6.83i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.70234818024636391223526054682, −10.10962054989421819802729445473, −9.390925786451992069911484911790, −8.167252697412986284206360444826, −7.57471185630051116697768219018, −6.76091717965462735401782781148, −4.91558855436626649322063352291, −4.40182772809741977455406200838, −2.74766722972222491960968597588, 0, 2.16397326081390733008067875029, 3.03911175838485880979269798732, 4.44798040693218445437951858890, 6.28764390657203301949820435716, 7.39205591932850735736184735306, 8.134024051869838672014470720806, 8.856973580495395662924411941078, 9.682676373863474969031114083293, 11.00254814791184934001903171770

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