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Properties

Label 2-360-72.11-c1-0-15
Degree $2$
Conductor $360$
Sign $-0.647 - 0.762i$
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  + (−0.0418 + 1.41i)2-s + (1.46 + 0.925i)3-s + (−1.99 − 0.118i)4-s + (0.5 + 0.866i)5-s + (−1.36 + 2.03i)6-s + (0.947 + 0.546i)7-s + (0.250 − 2.81i)8-s + (1.28 + 2.70i)9-s + (−1.24 + 0.670i)10-s + (2.31 + 1.33i)11-s + (−2.81 − 2.02i)12-s + (−2.12 + 1.22i)13-s + (−0.812 + 1.31i)14-s + (−0.0691 + 1.73i)15-s + (3.97 + 0.472i)16-s − 0.124i·17-s + ⋯
L(s)  = 1  + (−0.0295 + 0.999i)2-s + (0.845 + 0.534i)3-s + (−0.998 − 0.0591i)4-s + (0.223 + 0.387i)5-s + (−0.558 + 0.829i)6-s + (0.357 + 0.206i)7-s + (0.0886 − 0.996i)8-s + (0.429 + 0.903i)9-s + (−0.393 + 0.212i)10-s + (0.698 + 0.403i)11-s + (−0.812 − 0.583i)12-s + (−0.589 + 0.340i)13-s + (−0.217 + 0.351i)14-s + (−0.0178 + 0.446i)15-s + (0.993 + 0.118i)16-s − 0.0302i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 - 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.647 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.647 - 0.762i$
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: \((2,\ 360,\ (\ :1/2),\ -0.647 - 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.690397 + 1.49192i\)
\(L(\frac12)\) \(\approx\) \(0.690397 + 1.49192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0418 - 1.41i)T \)
3 \( 1 + (-1.46 - 0.925i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.947 - 0.546i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.31 - 1.33i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 0.124iT - 17T^{2} \)
19 \( 1 + 5.84T + 19T^{2} \)
23 \( 1 + (1.21 + 2.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.29 + 7.43i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.54 + 1.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.42iT - 37T^{2} \)
41 \( 1 + (-7.57 + 4.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.14 - 3.71i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.32 + 5.75i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.14T + 53T^{2} \)
59 \( 1 + (2.34 - 1.35i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.69 - 5.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.707 + 1.22i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 5.35T + 73T^{2} \)
79 \( 1 + (-2.07 - 1.19i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.2 - 6.47i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 + (-7.51 + 13.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.83566203561537142657160635874, −10.41662342134931157876148698541, −9.755797423346569847579247588777, −8.849612795825716856649145330904, −8.114306592914751382915457044658, −7.09401862537193652276227140304, −6.13818752734857095737663766925, −4.73004903639194539986287813380, −4.01276048490080552091170789382, −2.30716624194053746569898703935, 1.20041843256874950817633542973, 2.46000968284655251006583159800, 3.72239380461315981922178135680, 4.79619470388312815335925204913, 6.29281624389347669273823956167, 7.67253275454088806933949606603, 8.566905994130419307645271778139, 9.191145561476719529461899068393, 10.17194296525083343872657670660, 11.13204025877705635463562670729

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