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Properties

Label 2-360-24.11-c1-0-12
Degree $2$
Conductor $360$
Sign $0.954 + 0.297i$
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  + (1.38 − 0.305i)2-s + (1.81 − 0.844i)4-s + 5-s + 1.41i·7-s + (2.24 − 1.71i)8-s + (1.38 − 0.305i)10-s + 0.191i·11-s + 2.63i·13-s + (0.432 + 1.95i)14-s + (2.57 − 3.06i)16-s − 6.20i·17-s − 1.52·19-s + (1.81 − 0.844i)20-s + (0.0585 + 0.264i)22-s − 5.25·23-s + ⋯
L(s)  = 1  + (0.976 − 0.216i)2-s + (0.906 − 0.422i)4-s + 0.447·5-s + 0.534i·7-s + (0.793 − 0.608i)8-s + (0.436 − 0.0966i)10-s + 0.0577i·11-s + 0.731i·13-s + (0.115 + 0.521i)14-s + (0.643 − 0.765i)16-s − 1.50i·17-s − 0.349·19-s + (0.405 − 0.188i)20-s + (0.0124 + 0.0563i)22-s − 1.09·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.954 + 0.297i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.954 + 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.49888 - 0.379812i\)
\(L(\frac12)\) \(\approx\) \(2.49888 - 0.379812i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 + 0.305i)T \)
3 \( 1 \)
5 \( 1 - T \)
good7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 - 0.191iT - 11T^{2} \)
13 \( 1 - 2.63iT - 13T^{2} \)
17 \( 1 + 6.20iT - 17T^{2} \)
19 \( 1 + 1.52T + 19T^{2} \)
23 \( 1 + 5.25T + 23T^{2} \)
29 \( 1 + 0.270T + 29T^{2} \)
31 \( 1 - 6.20iT - 31T^{2} \)
37 \( 1 - 7.61iT - 37T^{2} \)
41 \( 1 + 9.22iT - 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 - 3.79T + 47T^{2} \)
53 \( 1 + 8.77T + 53T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + 0.382iT - 61T^{2} \)
67 \( 1 - 1.72T + 67T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 + 5.45T + 73T^{2} \)
79 \( 1 + 14.3iT - 79T^{2} \)
83 \( 1 - 15.2iT - 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 - 7.31T + 97T^{2} \)
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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

−11.82031008964013825224491873651, −10.59365165103862409789329916979, −9.748882949340482666328108286536, −8.719268643029259436445372635225, −7.28081878139800258974359656406, −6.42061013375936511300355554380, −5.40372622183108962916038069424, −4.49266654208795169986007175079, −3.06637737534298201299556997526, −1.88731179255881144931631660115, 1.94424931743962889361801291061, 3.45853119481712164088158435787, 4.45452071375734487487687775055, 5.74371697477974943568139040056, 6.38235637238582874938808205653, 7.62609889612532486040060671911, 8.389551016522769777151314106085, 9.960547709587365048001838022769, 10.65170523676494807374487109649, 11.57605519034687034773275674674

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