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Properties

Label 2-360-72.59-c1-0-7
Degree $2$
Conductor $360$
Sign $-0.0473 - 0.998i$
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  + (0.524 − 1.31i)2-s + (−0.0300 + 1.73i)3-s + (−1.45 − 1.37i)4-s + (−0.5 + 0.866i)5-s + (2.25 + 0.947i)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 + (2.42 − 2.46i)12-s + (4.44 + 2.56i)13-s + (0.807 + 5.57i)14-s + (−1.48 − 0.891i)15-s + (0.205 + 3.99i)16-s + 5.37i·17-s + ⋯
L(s)  = 1  + (0.370 − 0.928i)2-s + (−0.0173 + 0.999i)3-s + (−0.725 − 0.688i)4-s + (−0.223 + 0.387i)5-s + (0.922 + 0.386i)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 + (0.701 − 0.712i)12-s + (1.23 + 0.711i)13-s + (0.215 + 1.49i)14-s + (−0.383 − 0.230i)15-s + (0.0512 + 0.998i)16-s + 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0473 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0473 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0473 - 0.998i$
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: \((2,\ 360,\ (\ :1/2),\ -0.0473 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.560038 + 0.587242i\)
\(L(\frac12)\) \(\approx\) \(0.560038 + 0.587242i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.524 + 1.31i)T \)
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

−11.51266831663704813150366203441, −10.70617422584434249183004321299, −10.13333948010657439043721645906, −8.957069738561377008978380309431, −8.664072771230698075269288368393, −6.28270446453979641530154019916, −5.93054161380296427398421914303, −4.20145828406859619142201769098, −3.64852103895587204868214004715, −2.41187277540857275180486784795, 0.48172005579105087565809731502, 3.04635533385131976734192148179, 4.17695848951722531493114650419, 5.67809086546275520150315743970, 6.54511204630967127614927123501, 7.14584615113358138781968073620, 8.260178072216249021874042267480, 8.925897306821632172573606209369, 10.18068763370090084313453385800, 11.49746460597392442026528488940

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