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Properties

Label 2-360-72.11-c1-0-19
Degree $2$
Conductor $360$
Sign $0.814 + 0.579i$
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  + (0.913 − 1.07i)2-s + (−1.45 + 0.934i)3-s + (−0.329 − 1.97i)4-s + (0.5 + 0.866i)5-s + (−0.323 + 2.42i)6-s + (1.98 + 1.14i)7-s + (−2.42 − 1.44i)8-s + (1.25 − 2.72i)9-s + (1.39 + 0.251i)10-s + (4.17 + 2.40i)11-s + (2.32 + 2.56i)12-s + (1.25 − 0.723i)13-s + (3.05 − 1.09i)14-s + (−1.53 − 0.795i)15-s + (−3.78 + 1.29i)16-s − 5.75i·17-s + ⋯
L(s)  = 1  + (0.646 − 0.763i)2-s + (−0.841 + 0.539i)3-s + (−0.164 − 0.986i)4-s + (0.223 + 0.387i)5-s + (−0.132 + 0.991i)6-s + (0.750 + 0.433i)7-s + (−0.859 − 0.511i)8-s + (0.417 − 0.908i)9-s + (0.440 + 0.0796i)10-s + (1.25 + 0.726i)11-s + (0.670 + 0.741i)12-s + (0.347 − 0.200i)13-s + (0.815 − 0.292i)14-s + (−0.397 − 0.205i)15-s + (−0.945 + 0.324i)16-s − 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.814 + 0.579i$
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: \((2,\ 360,\ (\ :1/2),\ 0.814 + 0.579i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56943 - 0.501633i\)
\(L(\frac12)\) \(\approx\) \(1.56943 - 0.501633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.913 + 1.07i)T \)
3 \( 1 + (1.45 - 0.934i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-1.98 - 1.14i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.17 - 2.40i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.25 + 0.723i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.75iT - 17T^{2} \)
19 \( 1 - 5.54T + 19T^{2} \)
23 \( 1 + (0.878 + 1.52i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.85 - 6.67i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.38 + 1.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.48iT - 37T^{2} \)
41 \( 1 + (-5.84 + 3.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.513 + 0.889i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.87 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 14.2T + 53T^{2} \)
59 \( 1 + (7.06 - 4.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.27 - 1.31i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.258 + 0.448i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.60T + 71T^{2} \)
73 \( 1 - 1.59T + 73T^{2} \)
79 \( 1 + (-7.04 - 4.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.73 + 4.46i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.90iT - 89T^{2} \)
97 \( 1 + (-1.55 + 2.69i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.48182028773220870791875843178, −10.74322475886715714685867276481, −9.592934119054518879912192789198, −9.211542062054000745969643929031, −7.24438547381778177998311283135, −6.19817459811828221073073963773, −5.22996852163791859993214872009, −4.43771791507766303983415060522, −3.16107910498683098420018505979, −1.42289208817127060783242842847, 1.45818394040188533021074122400, 3.75762758354679356266171332863, 4.77468816349796294036615249775, 5.88643065905285719820430035675, 6.44489953836918023372855156997, 7.66671869566095302340324434423, 8.334723124159948218461430672991, 9.573462692718002262807473719213, 11.10195183154321371304706678497, 11.61931559858239127555191842404

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