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Properties

Label 2-360-72.11-c1-0-8
Degree $2$
Conductor $360$
Sign $-0.876 - 0.480i$
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  + (0.959 + 1.03i)2-s + (0.0478 + 1.73i)3-s + (−0.160 + 1.99i)4-s + (0.5 + 0.866i)5-s + (−1.75 + 1.71i)6-s + (−1.21 − 0.702i)7-s + (−2.22 + 1.74i)8-s + (−2.99 + 0.165i)9-s + (−0.420 + 1.35i)10-s + (1.54 + 0.889i)11-s + (−3.45 − 0.182i)12-s + (1.92 − 1.11i)13-s + (−0.436 − 1.93i)14-s + (−1.47 + 0.907i)15-s + (−3.94 − 0.639i)16-s − 2.44i·17-s + ⋯
L(s)  = 1  + (0.678 + 0.734i)2-s + (0.0276 + 0.999i)3-s + (−0.0802 + 0.996i)4-s + (0.223 + 0.387i)5-s + (−0.715 + 0.698i)6-s + (−0.459 − 0.265i)7-s + (−0.786 + 0.616i)8-s + (−0.998 + 0.0552i)9-s + (−0.132 + 0.426i)10-s + (0.464 + 0.268i)11-s + (−0.998 − 0.0526i)12-s + (0.535 − 0.308i)13-s + (−0.116 − 0.518i)14-s + (−0.380 + 0.234i)15-s + (−0.987 − 0.159i)16-s − 0.592i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.876 - 0.480i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.876 - 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.443506 + 1.73133i\)
\(L(\frac12)\) \(\approx\) \(0.443506 + 1.73133i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 1.03i)T \)
3 \( 1 + (-0.0478 - 1.73i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (1.21 + 0.702i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.54 - 0.889i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.92 + 1.11i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.44iT - 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
23 \( 1 + (-4.09 - 7.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.01 + 3.49i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.77 - 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 0.470iT - 37T^{2} \)
41 \( 1 + (-6.26 + 3.61i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.04 - 5.27i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.65 - 11.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 9.04T + 53T^{2} \)
59 \( 1 + (-8.51 + 4.91i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.96 + 2.28i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.54 + 2.68i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 1.45T + 73T^{2} \)
79 \( 1 + (7.90 + 4.56i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.76 - 3.32i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.82iT - 89T^{2} \)
97 \( 1 + (7.48 - 12.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.70410282480621255631104570395, −11.06958808775370217225248423627, −9.777688435635199162890839660897, −9.185469903754822801356808434186, −7.963700036618930931529560982514, −6.90868490843033861302968895742, −5.90711888314745528116653774106, −4.98559453478462530038028207792, −3.79754204462116553841243186834, −2.98483662890285110784239922415, 1.07930200547881533097972548009, 2.45153570190267094048225675058, 3.69643371797975367156147938493, 5.17805237917233675896023152616, 6.17676282660096247823858657461, 6.86614924530193673454648140519, 8.511735945573790654285577892902, 9.135057820682154978976442393178, 10.37352397904202308725338889631, 11.34501852431974042240125605774

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