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Properties

Label 2-360-72.59-c1-0-37
Degree $2$
Conductor $360$
Sign $0.727 - 0.685i$
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  + (1.39 + 0.202i)2-s + (−0.0300 + 1.73i)3-s + (1.91 + 0.567i)4-s + (0.5 − 0.866i)5-s + (−0.392 + 2.41i)6-s + (3.45 − 1.99i)7-s + (2.56 + 1.18i)8-s + (−2.99 − 0.104i)9-s + (0.875 − 1.11i)10-s + (0.520 − 0.300i)11-s + (−1.03 + 3.30i)12-s + (−4.44 − 2.56i)13-s + (5.23 − 2.08i)14-s + (1.48 + 0.891i)15-s + (3.35 + 2.17i)16-s + 5.37i·17-s + ⋯
L(s)  = 1  + (0.989 + 0.143i)2-s + (−0.0173 + 0.999i)3-s + (0.958 + 0.283i)4-s + (0.223 − 0.387i)5-s + (−0.160 + 0.987i)6-s + (1.30 − 0.753i)7-s + (0.908 + 0.417i)8-s + (−0.999 − 0.0347i)9-s + (0.276 − 0.351i)10-s + (0.157 − 0.0906i)11-s + (−0.300 + 0.953i)12-s + (−1.23 − 0.711i)13-s + (1.39 − 0.558i)14-s + (0.383 + 0.230i)15-s + (0.839 + 0.543i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.727 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.727 - 0.685i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.727 - 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40352 + 0.954192i\)
\(L(\frac12)\) \(\approx\) \(2.40352 + 0.954192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.202i)T \)
3 \( 1 + (0.0300 - 1.73i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-3.45 + 1.99i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.520 + 0.300i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.44 + 2.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.37iT - 17T^{2} \)
19 \( 1 + 8.29T + 19T^{2} \)
23 \( 1 + (-0.0667 + 0.115i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.507 - 0.879i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.58 + 0.914i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.87iT - 37T^{2} \)
41 \( 1 + (-4.43 - 2.56i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.36 + 5.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.27 + 9.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.25T + 53T^{2} \)
59 \( 1 + (-10.1 - 5.87i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.68 + 1.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.14 + 7.17i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 6.71T + 73T^{2} \)
79 \( 1 + (-7.14 + 4.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.85 - 1.65i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.54iT - 89T^{2} \)
97 \( 1 + (-2.33 - 4.04i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.50376407613956226554107104461, −10.65029231128853891822179015375, −10.16210514827494768494486336797, −8.519915174630615732763403031473, −7.912150344520020870555411008745, −6.49075946987698039512225654832, −5.29190990039618089927038335594, −4.61944159451024953843225773959, −3.79134692270657233620590705961, −2.11243014452616380348824720038, 1.94013350551967023242217028810, 2.57559789817741945805623817585, 4.52129252903680443398895770227, 5.40151400129521856500682409938, 6.50683075187130303838320218198, 7.28301471031896091291017475235, 8.251897195447101674203981549023, 9.512222107514354476742488706306, 11.02306418877540630168917193065, 11.46871218654621440065983382851

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