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Properties

Label 2-360-40.3-c1-0-24
Degree $2$
Conductor $360$
Sign $0.453 + 0.891i$
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  + (1.39 − 0.221i)2-s + (1.90 − 0.618i)4-s + (−1.17 − 1.90i)5-s + (−1.90 − 1.90i)7-s + (2.52 − 1.28i)8-s + (−2.06 − 2.39i)10-s + 3.23·11-s + (0.726 − 0.726i)13-s + (−3.07 − 2.23i)14-s + (3.23 − 2.35i)16-s + (1 − i)17-s + 2i·19-s + (−3.41 − 2.89i)20-s + (4.52 − 0.715i)22-s + (−4.25 + 4.25i)23-s + ⋯
L(s)  = 1  + (0.987 − 0.156i)2-s + (0.951 − 0.309i)4-s + (−0.525 − 0.850i)5-s + (−0.718 − 0.718i)7-s + (0.891 − 0.453i)8-s + (−0.652 − 0.757i)10-s + 0.975·11-s + (0.201 − 0.201i)13-s + (−0.822 − 0.597i)14-s + (0.809 − 0.587i)16-s + (0.242 − 0.242i)17-s + 0.458i·19-s + (−0.762 − 0.646i)20-s + (0.963 − 0.152i)22-s + (−0.886 + 0.886i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87624 - 1.14976i\)
\(L(\frac12)\) \(\approx\) \(1.87624 - 1.14976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.221i)T \)
3 \( 1 \)
5 \( 1 + (1.17 + 1.90i)T \)
good7 \( 1 + (1.90 + 1.90i)T + 7iT^{2} \)
11 \( 1 - 3.23T + 11T^{2} \)
13 \( 1 + (-0.726 + 0.726i)T - 13iT^{2} \)
17 \( 1 + (-1 + i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (4.25 - 4.25i)T - 23iT^{2} \)
29 \( 1 - 6.15T + 29T^{2} \)
31 \( 1 - 8.50iT - 31T^{2} \)
37 \( 1 + (0.726 + 0.726i)T + 37iT^{2} \)
41 \( 1 + 5.70T + 41T^{2} \)
43 \( 1 + (-4.61 - 4.61i)T + 43iT^{2} \)
47 \( 1 + (3.35 + 3.35i)T + 47iT^{2} \)
53 \( 1 + (-3.07 + 3.07i)T - 53iT^{2} \)
59 \( 1 + 0.472iT - 59T^{2} \)
61 \( 1 + 0.898iT - 61T^{2} \)
67 \( 1 + (4.61 - 4.61i)T - 67iT^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 + (4.70 + 4.70i)T + 73iT^{2} \)
79 \( 1 - 2.90T + 79T^{2} \)
83 \( 1 + (-6.61 - 6.61i)T + 83iT^{2} \)
89 \( 1 - 2.47iT - 89T^{2} \)
97 \( 1 + (-4.23 + 4.23i)T - 97iT^{2} \)
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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

−11.68584631817989744323690364211, −10.46445141514142685923671503649, −9.647784984097494717410003780588, −8.381551651687051648697445259076, −7.26418944681301344255249917958, −6.37769050494516020052855064330, −5.21499175878476153312290180087, −4.08905306128379260089780244976, −3.37553023258909505324731794737, −1.29489053719759326491388624404, 2.40561770836370361527521306456, 3.48751140858900777553901191744, 4.45179168663947462593306501699, 6.08367267687342379271236387599, 6.46970673509564982805339460436, 7.56710792587476057197104587336, 8.689874190828475624879335526169, 9.968821704898696644441623929929, 10.94157553944087414007041876278, 11.91366068146285058101380478081

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