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Properties

Label 2-360-24.11-c1-0-8
Degree $2$
Conductor $360$
Sign $-0.0381 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0381 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0381 + 0.999i$
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: \((2,\ 360,\ (\ :1/2),\ -0.0381 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471545 - 0.489891i\)
\(L(\frac12)\) \(\approx\) \(0.471545 - 0.489891i\)
\(L(\frac{3}{2})\) 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

−11.08465963664232259597760521013, −10.29858573375641786436050351359, −9.373286685741437464177548154427, −8.466710833981547157151772490688, −7.45499721792535724660743020827, −6.89425252288544559253851652305, −5.37638573831189931998924952720, −3.83524111825846302276724745974, −2.59070308067302866349665216807, −0.65039299344575439674815142797, 1.66626809346835669928442269973, 3.23740396396156240183684737019, 4.89351865727131501774244785814, 6.23514234043085971187256959208, 6.95827975598520280270052803775, 8.262869201625279357297394016928, 8.670402631112199777672128820850, 9.775487979158851297932688040122, 10.68153700578713028179883497690, 11.51646526526080049744770709027

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