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Properties

Label 2-360-360.77-c1-0-15
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (2.23 + 0.133i)5-s + (0.633 − 2.36i)6-s + (−0.767 + 2.86i)7-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + (−3.09 + 0.633i)10-s + (1 + 1.73i)11-s + 3.46i·12-s + (6.46 − 1.73i)13-s − 4.19i·14-s + (−2.13 + 3.23i)15-s + (1.99 − 3.46i)16-s + (−3 + 3i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (0.998 + 0.0599i)5-s + (0.258 − 0.965i)6-s + (−0.290 + 1.08i)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 + 0.999i·12-s + (1.79 − 0.480i)13-s − 1.12i·14-s + (−0.550 + 0.834i)15-s + (0.499 − 0.866i)16-s + (−0.727 + 0.727i)17-s + ⋯

Functional equation

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

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: $\chi_{360} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.460744 + 0.699851i\)
\(L(\frac12)\) \(\approx\) \(0.460744 + 0.699851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
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

−11.36183602088592094932999036061, −10.58701730159125283147090526399, −9.858428825795969717227450454290, −8.972190271443633507815261344918, −8.537433894834538864623849870231, −6.68123091850912379968081682033, −5.99253964535360062305549473134, −5.31956495790572260421940682472, −3.40554441284297033781978645065, −1.78185639619690260800517351482, 0.870115731589868472600962408350, 2.06699578117492395839295005303, 3.75227504656567874975225134702, 5.78363202728331103823639317830, 6.52743522837608655609495795542, 7.26842937255263520728776205640, 8.460760084699498731497810341785, 9.270084708931484413999817676490, 10.37680321832291065054379823664, 11.07198591118926751874409518798

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