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Properties

Label 2-360-24.11-c1-0-11
Degree $2$
Conductor $360$
Sign $0.816 + 0.577i$
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  + 1.41i·2-s − 2.00·4-s − 5-s − 4.24i·7-s − 2.82i·8-s − 1.41i·10-s − 1.41i·11-s − 4.24i·13-s + 6·14-s + 4.00·16-s + 2.82i·17-s − 4·19-s + 2.00·20-s + 2.00·22-s + 6·23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s − 0.447·5-s − 1.60i·7-s − 1.00i·8-s − 0.447i·10-s − 0.426i·11-s − 1.17i·13-s + 1.60·14-s + 1.00·16-s + 0.685i·17-s − 0.917·19-s + 0.447·20-s + 0.426·22-s + 1.25·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.816 + 0.577i$
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: \((2,\ 360,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848778 - 0.269773i\)
\(L(\frac12)\) \(\approx\) \(0.848778 - 0.269773i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 \)
5 \( 1 + T \)
good7 \( 1 + 4.24iT - 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 - 2.82iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + 1.41iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 8.48iT - 79T^{2} \)
83 \( 1 - 2.82iT - 83T^{2} \)
89 \( 1 - 7.07iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−11.00794501908516986370027919969, −10.50444401550395909437218691518, −9.365369084546064745117785379459, −8.189910695495045566734064790574, −7.60916422639482856644247854106, −6.71728405738356562238798562809, −5.60223145627249709613493317736, −4.33641379098073411985030648919, −3.54020984178657921228477008539, −0.63113860018792089149607210642, 1.92935022128585004072570941410, 3.03967244986734624438528594222, 4.46014190788157572885678232061, 5.34695776874268799749791666621, 6.72331570197606475065093664952, 8.141887114827673322842813735430, 9.122515437813939640607469354273, 9.437233547529379136720938173399, 10.97900678798030094794948149891, 11.45551312334139437742251812804

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