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Properties

Label 2-360-72.11-c1-0-47
Degree $2$
Conductor $360$
Sign $-0.731 + 0.682i$
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  + (1.26 − 0.641i)2-s + (−0.478 − 1.66i)3-s + (1.17 − 1.61i)4-s + (−0.5 − 0.866i)5-s + (−1.67 − 1.79i)6-s + (−1.88 − 1.09i)7-s + (0.446 − 2.79i)8-s + (−2.54 + 1.59i)9-s + (−1.18 − 0.770i)10-s + (2.33 + 1.34i)11-s + (−3.25 − 1.18i)12-s + (−1.45 + 0.842i)13-s + (−3.07 − 0.163i)14-s + (−1.20 + 1.24i)15-s + (−1.22 − 3.80i)16-s + 1.26i·17-s + ⋯
L(s)  = 1  + (0.891 − 0.453i)2-s + (−0.276 − 0.961i)3-s + (0.588 − 0.808i)4-s + (−0.223 − 0.387i)5-s + (−0.682 − 0.731i)6-s + (−0.713 − 0.412i)7-s + (0.158 − 0.987i)8-s + (−0.847 + 0.531i)9-s + (−0.374 − 0.243i)10-s + (0.703 + 0.406i)11-s + (−0.939 − 0.342i)12-s + (−0.404 + 0.233i)13-s + (−0.823 − 0.0435i)14-s + (−0.310 + 0.321i)15-s + (−0.306 − 0.951i)16-s + 0.306i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.731 + 0.682i$
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: \((2,\ 360,\ (\ :1/2),\ -0.731 + 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656024 - 1.66527i\)
\(L(\frac12)\) \(\approx\) \(0.656024 - 1.66527i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.641i)T \)
3 \( 1 + (0.478 + 1.66i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1.88 + 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.33 - 1.34i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.45 - 0.842i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.26iT - 17T^{2} \)
19 \( 1 - 7.49T + 19T^{2} \)
23 \( 1 + (3.13 + 5.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.81 + 3.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.27 + 3.62i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.85iT - 37T^{2} \)
41 \( 1 + (3.69 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.78 + 10.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.33 - 5.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.537T + 53T^{2} \)
59 \( 1 + (4.48 - 2.58i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.13 - 5.27i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.42 - 7.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.23T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + (-9.75 - 5.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.764 + 0.441i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.4iT - 89T^{2} \)
97 \( 1 + (4.16 - 7.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.65774958347383842061378983637, −10.30608865524837183208043638470, −9.514505430419145406630489759357, −8.039803156113907010604067997318, −6.93923942457738527883509075178, −6.31535565545337786036270873553, −5.12945400088154125259949671606, −3.96181594139183434186442779092, −2.58248031975876392037886499108, −1.02090353786395213081911792669, 3.00564124837737726816439249368, 3.64460715076954952584264584371, 4.97734491450226387224521810962, 5.83182513793358073389369603708, 6.74312429861963636507567020175, 7.88517709958799541579604386248, 9.146855345944007256160086486884, 9.928644141132384470984876006203, 11.17043331720270410411047458059, 11.79224650627603524370984187828

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