Properties

Label 2-360-72.59-c1-0-41
Degree $2$
Conductor $360$
Sign $0.941 + 0.335i$
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  + (1.29 + 0.565i)2-s + (0.284 − 1.70i)3-s + (1.35 + 1.46i)4-s + (0.5 − 0.866i)5-s + (1.33 − 2.05i)6-s + (2.24 − 1.29i)7-s + (0.933 + 2.67i)8-s + (−2.83 − 0.970i)9-s + (1.13 − 0.839i)10-s + (−3.78 + 2.18i)11-s + (2.89 − 1.90i)12-s + (2.94 + 1.69i)13-s + (3.64 − 0.410i)14-s + (−1.33 − 1.10i)15-s + (−0.301 + 3.98i)16-s − 8.12i·17-s + ⋯
L(s)  = 1  + (0.916 + 0.400i)2-s + (0.163 − 0.986i)3-s + (0.679 + 0.733i)4-s + (0.223 − 0.387i)5-s + (0.544 − 0.838i)6-s + (0.849 − 0.490i)7-s + (0.329 + 0.944i)8-s + (−0.946 − 0.323i)9-s + (0.359 − 0.265i)10-s + (−1.14 + 0.658i)11-s + (0.834 − 0.550i)12-s + (0.816 + 0.471i)13-s + (0.974 − 0.109i)14-s + (−0.345 − 0.284i)15-s + (−0.0753 + 0.997i)16-s − 1.96i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.941 + 0.335i$
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: \((2,\ 360,\ (\ :1/2),\ 0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.45385 - 0.424152i\)
\(L(\frac12)\) \(\approx\) \(2.45385 - 0.424152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.565i)T \)
3 \( 1 + (-0.284 + 1.70i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (-2.24 + 1.29i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.78 - 2.18i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.94 - 1.69i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 8.12iT - 17T^{2} \)
19 \( 1 - 0.0767T + 19T^{2} \)
23 \( 1 + (1.81 - 3.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.63 - 8.03i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.23 + 3.60i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.03iT - 37T^{2} \)
41 \( 1 + (-0.999 - 0.577i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.09 - 5.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.608 + 1.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + (12.8 + 7.40i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.8 + 6.23i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.0137 + 0.0237i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.518T + 71T^{2} \)
73 \( 1 - 1.51T + 73T^{2} \)
79 \( 1 + (-0.0609 + 0.0351i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.39 - 4.84i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.30iT - 89T^{2} \)
97 \( 1 + (-0.938 - 1.62i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.56079099063700048762064324453, −10.96464795193550951476895011982, −9.335229278497837436594958195613, −8.118501137671730911013003956611, −7.53075666192689563244736663891, −6.68400826241882597964753175994, −5.41056369271325633803445091556, −4.67955036919774223242168915081, −3.02975922491266520990742727737, −1.70366636584354550720549565033, 2.18193811693162326498729564259, 3.35688089571332915819652315975, 4.40006552316423054169072119359, 5.60487998212823463435249422204, 6.04005362854710777349620575988, 7.941920051455992142261420884483, 8.684621978091862848650833783972, 10.16573349720918131124122056267, 10.70924415251972207082639770398, 11.20385313798461037183877108745

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