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Properties

Label 2-360-72.11-c1-0-27
Degree $2$
Conductor $360$
Sign $0.630 - 0.776i$
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  + (0.951 + 1.04i)2-s + (1.28 − 1.16i)3-s + (−0.191 + 1.99i)4-s + (0.5 + 0.866i)5-s + (2.43 + 0.242i)6-s + (1.13 + 0.653i)7-s + (−2.26 + 1.69i)8-s + (0.306 − 2.98i)9-s + (−0.430 + 1.34i)10-s + (0.447 + 0.258i)11-s + (2.06 + 2.78i)12-s + (1.21 − 0.701i)13-s + (0.392 + 1.80i)14-s + (1.64 + 0.533i)15-s + (−3.92 − 0.760i)16-s + 2.63i·17-s + ⋯
L(s)  = 1  + (0.672 + 0.740i)2-s + (0.742 − 0.670i)3-s + (−0.0955 + 0.995i)4-s + (0.223 + 0.387i)5-s + (0.995 + 0.0988i)6-s + (0.427 + 0.246i)7-s + (−0.800 + 0.598i)8-s + (0.102 − 0.994i)9-s + (−0.136 + 0.425i)10-s + (0.134 + 0.0778i)11-s + (0.596 + 0.802i)12-s + (0.337 − 0.194i)13-s + (0.104 + 0.482i)14-s + (0.425 + 0.137i)15-s + (−0.981 − 0.190i)16-s + 0.638i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.630 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.630 - 0.776i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.630 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17818 + 1.03688i\)
\(L(\frac12)\) \(\approx\) \(2.17818 + 1.03688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 1.04i)T \)
3 \( 1 + (-1.28 + 1.16i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.13 - 0.653i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.447 - 0.258i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.21 + 0.701i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.63iT - 17T^{2} \)
19 \( 1 + 2.33T + 19T^{2} \)
23 \( 1 + (2.91 + 5.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.53 - 2.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.36 + 4.82i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.68iT - 37T^{2} \)
41 \( 1 + (2.75 - 1.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.15 + 2.00i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.479 + 0.830i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.64T + 53T^{2} \)
59 \( 1 + (7.90 - 4.56i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.31 + 4.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.18 + 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.93T + 71T^{2} \)
73 \( 1 - 7.66T + 73T^{2} \)
79 \( 1 + (-13.5 - 7.82i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.39 - 4.84i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.47iT - 89T^{2} \)
97 \( 1 + (2.47 - 4.29i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.98541740377649032769511111948, −10.78121917344004269668846980405, −9.417048217259526523274163273794, −8.345003778219758267476302960091, −7.898458808656928532263269320618, −6.61648839140689708211819583537, −6.10905348407627347120298176497, −4.58882758865750087902000280600, −3.37865977194966903426816188574, −2.15428073447137676230296628774, 1.69625140759490303762048142719, 3.06885247635809976589221505496, 4.22489416004123683006597400992, 4.97036462990556062682583828321, 6.17752215669815310304842359499, 7.72198419569125015706625259889, 8.849151753339792008380931535444, 9.591190266847329487766319211166, 10.43460043110712937036924831280, 11.27422691500653153577310765990

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