Properties

Label 2-360-1.1-c7-0-8
Degree $2$
Conductor $360$
Sign $1$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 125·5-s + 776·7-s + 124·11-s − 1.30e4·13-s + 1.59e4·17-s − 2.05e4·19-s + 2.92e4·23-s + 1.56e4·25-s + 2.21e5·29-s − 1.09e5·31-s − 9.70e4·35-s + 7.34e4·37-s − 1.27e4·41-s + 2.90e5·43-s − 1.26e6·47-s − 2.21e5·49-s + 3.95e5·53-s − 1.55e4·55-s − 4.21e5·59-s − 2.12e6·61-s + 1.63e6·65-s − 3.13e6·67-s + 5.37e6·71-s + 4.98e6·73-s + 9.62e4·77-s + 3.86e6·79-s + 6.19e6·83-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.855·7-s + 0.0280·11-s − 1.65·13-s + 0.787·17-s − 0.686·19-s + 0.500·23-s + 1/5·25-s + 1.68·29-s − 0.661·31-s − 0.382·35-s + 0.238·37-s − 0.0289·41-s + 0.557·43-s − 1.78·47-s − 0.268·49-s + 0.365·53-s − 0.0125·55-s − 0.267·59-s − 1.19·61-s + 0.738·65-s − 1.27·67-s + 1.78·71-s + 1.49·73-s + 0.0240·77-s + 0.882·79-s + 1.18·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $1$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.864308163\)
\(L(\frac12)\) \(\approx\) \(1.864308163\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + p^{3} T \)
good7 \( 1 - 776 T + p^{7} T^{2} \)
11 \( 1 - 124 T + p^{7} T^{2} \)
13 \( 1 + 13082 T + p^{7} T^{2} \)
17 \( 1 - 15950 T + p^{7} T^{2} \)
19 \( 1 + 20516 T + p^{7} T^{2} \)
23 \( 1 - 29224 T + p^{7} T^{2} \)
29 \( 1 - 221482 T + p^{7} T^{2} \)
31 \( 1 + 109760 T + p^{7} T^{2} \)
37 \( 1 - 73422 T + p^{7} T^{2} \)
41 \( 1 + 12762 T + p^{7} T^{2} \)
43 \( 1 - 290548 T + p^{7} T^{2} \)
47 \( 1 + 1269152 T + p^{7} T^{2} \)
53 \( 1 - 395778 T + p^{7} T^{2} \)
59 \( 1 + 421492 T + p^{7} T^{2} \)
61 \( 1 + 2122250 T + p^{7} T^{2} \)
67 \( 1 + 3132868 T + p^{7} T^{2} \)
71 \( 1 - 5376552 T + p^{7} T^{2} \)
73 \( 1 - 4985466 T + p^{7} T^{2} \)
79 \( 1 - 3867504 T + p^{7} T^{2} \)
83 \( 1 - 6190196 T + p^{7} T^{2} \)
89 \( 1 + 1124394 T + p^{7} T^{2} \)
97 \( 1 - 9968098 T + p^{7} T^{2} \)
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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

−10.29614537354276201813131913619, −9.347631868800501618983302008718, −8.204193832575667780360067782496, −7.58894631378346946643437975869, −6.55169416872650730496791710750, −5.12945754143845893777011604602, −4.53383300594311785606830292203, −3.13101772480386563946661555509, −1.96811343419952326424001396063, −0.63822612443783554148258708191, 0.63822612443783554148258708191, 1.96811343419952326424001396063, 3.13101772480386563946661555509, 4.53383300594311785606830292203, 5.12945754143845893777011604602, 6.55169416872650730496791710750, 7.58894631378346946643437975869, 8.204193832575667780360067782496, 9.347631868800501618983302008718, 10.29614537354276201813131913619

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