Properties

Label 2-360-5.2-c2-0-6
Degree $2$
Conductor $360$
Sign $0.945 - 0.326i$
Analytic cond. $9.80928$
Root an. cond. $3.13197$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.67 + 4.22i)5-s + (−5.44 − 5.44i)7-s + 6.44·11-s + (14.4 − 14.4i)13-s + (23.1 + 23.1i)17-s + 16.6i·19-s + (6.65 − 6.65i)23-s + (−10.6 + 22.5i)25-s + 0.0454i·29-s + 4.49·31-s + (8.44 − 37.5i)35-s + (35.3 + 35.3i)37-s − 20.2·41-s + (32.2 − 32.2i)43-s + (50.5 + 50.5i)47-s + ⋯
L(s)  = 1  + (0.534 + 0.844i)5-s + (−0.778 − 0.778i)7-s + 0.586·11-s + (1.11 − 1.11i)13-s + (1.36 + 1.36i)17-s + 0.878i·19-s + (0.289 − 0.289i)23-s + (−0.427 + 0.903i)25-s + 0.00156i·29-s + 0.144·31-s + (0.241 − 1.07i)35-s + (0.955 + 0.955i)37-s − 0.494·41-s + (0.750 − 0.750i)43-s + (1.07 + 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(9.80928\)
Root analytic conductor: \(3.13197\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1),\ 0.945 - 0.326i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.83292 + 0.307565i\)
\(L(\frac12)\) \(\approx\) \(1.83292 + 0.307565i\)
\(L(2)\) 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 \( 1 + (-2.67 - 4.22i)T \)
good7 \( 1 + (5.44 + 5.44i)T + 49iT^{2} \)
11 \( 1 - 6.44T + 121T^{2} \)
13 \( 1 + (-14.4 + 14.4i)T - 169iT^{2} \)
17 \( 1 + (-23.1 - 23.1i)T + 289iT^{2} \)
19 \( 1 - 16.6iT - 361T^{2} \)
23 \( 1 + (-6.65 + 6.65i)T - 529iT^{2} \)
29 \( 1 - 0.0454iT - 841T^{2} \)
31 \( 1 - 4.49T + 961T^{2} \)
37 \( 1 + (-35.3 - 35.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 20.2T + 1.68e3T^{2} \)
43 \( 1 + (-32.2 + 32.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (-50.5 - 50.5i)T + 2.20e3iT^{2} \)
53 \( 1 + (-5.50 + 5.50i)T - 2.80e3iT^{2} \)
59 \( 1 + 55.4iT - 3.48e3T^{2} \)
61 \( 1 - 47.8T + 3.72e3T^{2} \)
67 \( 1 + (85.2 + 85.2i)T + 4.48e3iT^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (21.9 - 21.9i)T - 5.32e3iT^{2} \)
79 \( 1 + 126. iT - 6.24e3T^{2} \)
83 \( 1 + (94.9 - 94.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 71.7iT - 7.92e3T^{2} \)
97 \( 1 + (37 + 37i)T + 9.40e3iT^{2} \)
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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

−10.95757136315232717224371810510, −10.34145733431362358169182812214, −9.760172016180964781043358253942, −8.397680839023342874267954751595, −7.44665179369290113240234190230, −6.28624544422927571632522958166, −5.81283546576872512495056068151, −3.85546864239644418131633959357, −3.16333299224535803071774159056, −1.26016379117886520388816739780, 1.09779059462144061013215806981, 2.70534638790629915063946468381, 4.14445492835813815464481426349, 5.42410133605028082534081492524, 6.19720245700129935888390956868, 7.30092751217430416349477067918, 8.854456701010878744497318267836, 9.136160949045945125723591998298, 9.977884706798378282113167369976, 11.44834575612007930245459110526

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