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Properties

Label 2-360-72.59-c1-0-33
Degree $2$
Conductor $360$
Sign $0.698 + 0.715i$
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  + (−1.17 − 0.793i)2-s + (1.72 + 0.199i)3-s + (0.739 + 1.85i)4-s + (0.5 − 0.866i)5-s + (−1.85 − 1.59i)6-s + (3.97 − 2.29i)7-s + (0.609 − 2.76i)8-s + (2.92 + 0.686i)9-s + (−1.27 + 0.616i)10-s + (−4.08 + 2.36i)11-s + (0.901 + 3.34i)12-s + (−1.87 − 1.08i)13-s + (−6.46 − 0.469i)14-s + (1.03 − 1.39i)15-s + (−2.90 + 2.74i)16-s + 1.23i·17-s + ⋯
L(s)  = 1  + (−0.827 − 0.561i)2-s + (0.993 + 0.115i)3-s + (0.369 + 0.929i)4-s + (0.223 − 0.387i)5-s + (−0.757 − 0.652i)6-s + (1.50 − 0.866i)7-s + (0.215 − 0.976i)8-s + (0.973 + 0.228i)9-s + (−0.402 + 0.194i)10-s + (−1.23 + 0.711i)11-s + (0.260 + 0.965i)12-s + (−0.518 − 0.299i)13-s + (−1.72 − 0.125i)14-s + (0.266 − 0.358i)15-s + (−0.726 + 0.687i)16-s + 0.299i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.698 + 0.715i$
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: \((2,\ 360,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33137 - 0.560650i\)
\(L(\frac12)\) \(\approx\) \(1.33137 - 0.560650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.793i)T \)
3 \( 1 + (-1.72 - 0.199i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-3.97 + 2.29i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.08 - 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.87 + 1.08i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.23iT - 17T^{2} \)
19 \( 1 - 4.35T + 19T^{2} \)
23 \( 1 + (0.117 - 0.204i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.84 + 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.06 + 2.34i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.91iT - 37T^{2} \)
41 \( 1 + (-6.34 - 3.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.62 + 4.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.05 + 1.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.92T + 53T^{2} \)
59 \( 1 + (10.1 + 5.88i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.27 - 3.62i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.51 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.851T + 71T^{2} \)
73 \( 1 - 10.6T + 73T^{2} \)
79 \( 1 + (10.0 - 5.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.35 - 1.35i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.9iT - 89T^{2} \)
97 \( 1 + (0.816 + 1.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.06964558937331821036221877373, −10.20184525581094659375152524392, −9.643964199953983177286170368997, −8.430432916832189970557479933898, −7.75869404055609060002599098853, −7.33045004442470093318470636290, −5.05599084690438890995433250195, −4.09902496585508753164939194365, −2.58067393144251494100841081072, −1.46512711185881609782132134862, 1.78606626862635486991145418284, 2.81114306185383319033595650106, 4.93120687789432825759607395400, 5.75140586299210010056355744168, 7.40288840554026392547355781064, 7.73125749112454121792081644352, 8.780627519415115034150376311106, 9.327507095199030589141252313701, 10.56247778129700507346861513673, 11.20861240984849985946547340196

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