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Properties

Label 2-360-72.59-c1-0-18
Degree $2$
Conductor $360$
Sign $0.932 + 0.361i$
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  + (−0.325 − 1.37i)2-s + (1.62 + 0.609i)3-s + (−1.78 + 0.897i)4-s + (−0.5 + 0.866i)5-s + (0.310 − 2.42i)6-s + (0.518 − 0.299i)7-s + (1.81 + 2.16i)8-s + (2.25 + 1.97i)9-s + (1.35 + 0.405i)10-s + (0.700 − 0.404i)11-s + (−3.44 + 0.365i)12-s + (3.52 + 2.03i)13-s + (−0.581 − 0.616i)14-s + (−1.33 + 1.09i)15-s + (2.39 − 3.20i)16-s − 1.87i·17-s + ⋯
L(s)  = 1  + (−0.230 − 0.973i)2-s + (0.936 + 0.351i)3-s + (−0.893 + 0.448i)4-s + (−0.223 + 0.387i)5-s + (0.126 − 0.991i)6-s + (0.196 − 0.113i)7-s + (0.642 + 0.766i)8-s + (0.752 + 0.658i)9-s + (0.428 + 0.128i)10-s + (0.211 − 0.121i)11-s + (−0.994 + 0.105i)12-s + (0.977 + 0.564i)13-s + (−0.155 − 0.164i)14-s + (−0.345 + 0.283i)15-s + (0.597 − 0.801i)16-s − 0.453i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.932 + 0.361i$
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: \((2,\ 360,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51670 - 0.284049i\)
\(L(\frac12)\) \(\approx\) \(1.51670 - 0.284049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.325 + 1.37i)T \)
3 \( 1 + (-1.62 - 0.609i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.518 + 0.299i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.700 + 0.404i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.52 - 2.03i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.87iT - 17T^{2} \)
19 \( 1 - 3.32T + 19T^{2} \)
23 \( 1 + (1.00 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.38 + 2.40i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.04 - 2.33i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.30iT - 37T^{2} \)
41 \( 1 + (7.32 + 4.22i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.19 + 7.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.76 + 11.7i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.27T + 53T^{2} \)
59 \( 1 + (-6.41 - 3.70i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.38 + 2.53i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.68 + 11.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.24T + 71T^{2} \)
73 \( 1 + 16.0T + 73T^{2} \)
79 \( 1 + (3.85 - 2.22i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.04 + 2.33i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.41iT - 89T^{2} \)
97 \( 1 + (-1.95 - 3.38i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.37658036291090957068515340413, −10.36571156758379943970946481909, −9.666094713441322110692863018106, −8.708559808125510594437919331139, −8.043513471199369290062140504731, −6.89051947258435952644566374920, −5.05236005512475747860932969579, −3.88510441516769051219100199398, −3.12086986986284411965682219605, −1.67978164360023059765240320249, 1.33121887772036094946076281792, 3.41618663581810432651619678683, 4.55946098186348233306933933592, 5.87892471921640316140199073467, 6.88703598101594028403201897502, 7.989291550546694082238755191094, 8.396715668942409212521552336024, 9.342152176659475002352879225441, 10.18398436210365148525286008232, 11.55186512378826511339133809073

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