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Properties

Label 2-360-8.5-c5-0-93
Degree $2$
Conductor $360$
Sign $-0.993 + 0.116i$
Analytic cond. $57.7381$
Root an. cond. $7.59856$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.78 − 3.01i)2-s + (13.7 − 28.8i)4-s + 25i·5-s − 56.4·7-s + (−21.0 − 179. i)8-s + (75.4 + 119. i)10-s − 261. i·11-s − 720. i·13-s + (−270. + 170. i)14-s + (−643. − 796. i)16-s + 1.87e3·17-s + 1.99e3i·19-s + (721. + 344. i)20-s + (−787. − 1.24e3i)22-s − 2.57e3·23-s + ⋯
L(s)  = 1  + (0.845 − 0.533i)2-s + (0.431 − 0.902i)4-s + 0.447i·5-s − 0.435·7-s + (−0.116 − 0.993i)8-s + (0.238 + 0.378i)10-s − 0.650i·11-s − 1.18i·13-s + (−0.368 + 0.232i)14-s + (−0.628 − 0.778i)16-s + 1.57·17-s + 1.26i·19-s + (0.403 + 0.192i)20-s + (−0.346 − 0.550i)22-s − 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.993 + 0.116i$
Analytic conductor: \(57.7381\)
Root analytic conductor: \(7.59856\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :5/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.893789390\)
\(L(\frac12)\) \(\approx\) \(1.893789390\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.78 + 3.01i)T \)
3 \( 1 \)
5 \( 1 - 25iT \)
good7 \( 1 + 56.4T + 1.68e4T^{2} \)
11 \( 1 + 261. iT - 1.61e5T^{2} \)
13 \( 1 + 720. iT - 3.71e5T^{2} \)
17 \( 1 - 1.87e3T + 1.41e6T^{2} \)
19 \( 1 - 1.99e3iT - 2.47e6T^{2} \)
23 \( 1 + 2.57e3T + 6.43e6T^{2} \)
29 \( 1 - 1.70e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.73e3T + 2.86e7T^{2} \)
37 \( 1 + 1.22e4iT - 6.93e7T^{2} \)
41 \( 1 + 1.49e4T + 1.15e8T^{2} \)
43 \( 1 + 1.81e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.14e3T + 2.29e8T^{2} \)
53 \( 1 - 1.60e3iT - 4.18e8T^{2} \)
59 \( 1 - 2.68e3iT - 7.14e8T^{2} \)
61 \( 1 + 4.45e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.24e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.18e3T + 1.80e9T^{2} \)
73 \( 1 + 4.10e4T + 2.07e9T^{2} \)
79 \( 1 - 4.63e4T + 3.07e9T^{2} \)
83 \( 1 - 6.16e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.32e4T + 5.58e9T^{2} \)
97 \( 1 + 3.92e4T + 8.58e9T^{2} \)
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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

−10.31729815656893805297178151507, −9.707208163288533357170667149913, −8.195665420382237564243755648096, −7.17183890756217614105776189799, −5.85121668201411534462030908350, −5.49150911436203420454280454933, −3.67652534147793455630545132513, −3.25650353783891998038891973743, −1.77584305025505061400310945511, −0.32724985338002001614988813868, 1.71334156750915753822295096510, 3.12421158080268826447540289508, 4.26699966708305401168219599983, 5.13219339965880535620367483661, 6.24645887524989864403480775960, 7.11495678030438707807720204463, 8.041244713476131031367194288282, 9.147102279131487418021280779502, 10.02888605630489800997346391492, 11.49278957330199262281645114410

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