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Properties

Label 2-360-45.4-c1-0-3
Degree $2$
Conductor $360$
Sign $0.662 - 0.749i$
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  + (−1.56 + 0.750i)3-s + (−2.18 − 0.497i)5-s + (2.51 − 1.45i)7-s + (1.87 − 2.34i)9-s + (2.96 + 5.14i)11-s + (−1.91 − 1.10i)13-s + (3.77 − 0.860i)15-s + 7.22i·17-s + 1.89·19-s + (−2.83 + 4.15i)21-s + (7.15 + 4.13i)23-s + (4.50 + 2.16i)25-s + (−1.16 + 5.06i)27-s + (−2.08 − 3.61i)29-s + (−0.617 + 1.06i)31-s + ⋯
L(s)  = 1  + (−0.901 + 0.433i)3-s + (−0.974 − 0.222i)5-s + (0.951 − 0.549i)7-s + (0.624 − 0.781i)9-s + (0.894 + 1.55i)11-s + (−0.532 − 0.307i)13-s + (0.975 − 0.222i)15-s + 1.75i·17-s + 0.434·19-s + (−0.619 + 0.907i)21-s + (1.49 + 0.861i)23-s + (0.900 + 0.433i)25-s + (−0.223 + 0.974i)27-s + (−0.387 − 0.670i)29-s + (−0.110 + 0.192i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862541 + 0.388802i\)
\(L(\frac12)\) \(\approx\) \(0.862541 + 0.388802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.56 - 0.750i)T \)
5 \( 1 + (2.18 + 0.497i)T \)
good7 \( 1 + (-2.51 + 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.96 - 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.91 + 1.10i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.22iT - 17T^{2} \)
19 \( 1 - 1.89T + 19T^{2} \)
23 \( 1 + (-7.15 - 4.13i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.08 + 3.61i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.617 - 1.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.466iT - 37T^{2} \)
41 \( 1 + (-4.84 + 8.39i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.29 + 2.47i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.183 + 0.105i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.33iT - 53T^{2} \)
59 \( 1 + (2.02 - 3.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.85 - 8.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.62 - 4.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.70T + 71T^{2} \)
73 \( 1 + 1.17iT - 73T^{2} \)
79 \( 1 + (4.36 + 7.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 - 6.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-4.91 + 2.83i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.50528227578937494403245031112, −10.81917861983312710955205482459, −9.910900750646144278835786261222, −8.849040674865226284090060503409, −7.53763960733141633259056424891, −7.03310597021283836271617192821, −5.49703572813653685118402813827, −4.46940614989491504387483070037, −3.90221922867036168645935602849, −1.36326031550375313028113959833, 0.885309028420996486895299000564, 2.91717710309873735596194185518, 4.54365661543054483611655661113, 5.35905503883977777972167151490, 6.62510142243054227796028462053, 7.42252908369487383598655908840, 8.407953918427578361873684416908, 9.362749559632039709071644741185, 11.10420412728203853954129978026, 11.24543768202317315967514256118

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