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Properties

Label 2-360-72.59-c1-0-8
Degree $2$
Conductor $360$
Sign $-0.911 - 0.411i$
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 + 0.204i)2-s + (0.875 + 1.49i)3-s + (1.91 − 0.573i)4-s + (−0.5 + 0.866i)5-s + (−1.53 − 1.91i)6-s + (−1.05 + 0.608i)7-s + (−2.56 + 1.19i)8-s + (−1.46 + 2.61i)9-s + (0.522 − 1.31i)10-s + (−2.23 + 1.29i)11-s + (2.53 + 2.36i)12-s + (−4.16 − 2.40i)13-s + (1.34 − 1.06i)14-s + (−1.73 + 0.0107i)15-s + (3.34 − 2.19i)16-s + 5.97i·17-s + ⋯
L(s)  = 1  + (−0.989 + 0.144i)2-s + (0.505 + 0.862i)3-s + (0.958 − 0.286i)4-s + (−0.223 + 0.387i)5-s + (−0.625 − 0.780i)6-s + (−0.398 + 0.229i)7-s + (−0.906 + 0.422i)8-s + (−0.489 + 0.872i)9-s + (0.165 − 0.415i)10-s + (−0.673 + 0.389i)11-s + (0.731 + 0.681i)12-s + (−1.15 − 0.667i)13-s + (0.360 − 0.285i)14-s + (−0.447 + 0.00278i)15-s + (0.835 − 0.549i)16-s + 1.44i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.911 - 0.411i$
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: \((2,\ 360,\ (\ :1/2),\ -0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.134471 + 0.624768i\)
\(L(\frac12)\) \(\approx\) \(0.134471 + 0.624768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.204i)T \)
3 \( 1 + (-0.875 - 1.49i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (1.05 - 0.608i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.23 - 1.29i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.16 + 2.40i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 5.97iT - 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 + (0.866 - 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.85 + 6.67i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.30 - 2.48i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.33iT - 37T^{2} \)
41 \( 1 + (2.00 + 1.16i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.97 + 3.41i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.25 - 5.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.0729T + 53T^{2} \)
59 \( 1 + (-5.32 - 3.07i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.89 - 5.71i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.59 + 4.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.50T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + (-9.17 + 5.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.02 + 2.90i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 + (-4.84 - 8.39i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.55418926710070867848039447312, −10.41113806761461119748494989546, −10.06350236371842235731586708620, −9.235212376597598927717455794088, −8.032598959853730315917297062874, −7.60397958695661695905296439472, −6.16127874724875812854447197505, −5.03993641008441958612710611056, −3.37881379136413953107252471300, −2.36627203125984951531078046471, 0.52742383035483205803202953160, 2.26204732772845716877629573982, 3.34090704999365734859531437809, 5.29600709730843200228082991230, 6.78954084445086358692690038653, 7.37842506179264222589416779923, 8.181126153047535714907509947703, 9.279045892205488667144179805167, 9.725283611140203736667009222688, 11.12996765689410453653136096951

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