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Properties

Label 2-360-9.4-c1-0-0
Degree $2$
Conductor $360$
Sign $-0.964 - 0.265i$
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  + (−1.73 + 0.0696i)3-s + (−0.5 + 0.866i)5-s + (0.165 + 0.287i)7-s + (2.99 − 0.240i)9-s + (−2.20 − 3.81i)11-s + (−3.49 + 6.04i)13-s + (0.805 − 1.53i)15-s − 3.07·17-s − 7.55·19-s + (−0.307 − 0.485i)21-s + (−0.834 + 1.44i)23-s + (−0.499 − 0.866i)25-s + (−5.15 + 0.625i)27-s + (−1.78 − 3.09i)29-s + (−2.82 + 4.88i)31-s + ⋯
L(s)  = 1  + (−0.999 + 0.0401i)3-s + (−0.223 + 0.387i)5-s + (0.0627 + 0.108i)7-s + (0.996 − 0.0803i)9-s + (−0.664 − 1.15i)11-s + (−0.968 + 1.67i)13-s + (0.207 − 0.395i)15-s − 0.744·17-s − 1.73·19-s + (−0.0670 − 0.106i)21-s + (−0.173 + 0.301i)23-s + (−0.0999 − 0.173i)25-s + (−0.992 + 0.120i)27-s + (−0.331 − 0.574i)29-s + (−0.506 + 0.877i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.964 - 0.265i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0285847 + 0.211505i\)
\(L(\frac12)\) \(\approx\) \(0.0285847 + 0.211505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.73 - 0.0696i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-0.165 - 0.287i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.20 + 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.49 - 6.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.07T + 17T^{2} \)
19 \( 1 + 7.55T + 19T^{2} \)
23 \( 1 + (0.834 - 1.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.78 + 3.09i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.82 - 4.88i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.64T + 37T^{2} \)
41 \( 1 + (2.74 - 4.76i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.53 - 6.12i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.98 + 6.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.47T + 53T^{2} \)
59 \( 1 + (-3.69 + 6.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.296 + 0.513i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.61 - 4.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.98T + 71T^{2} \)
73 \( 1 + 8.05T + 73T^{2} \)
79 \( 1 + (-5.49 - 9.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.14 - 5.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-0.622 - 1.07i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.52425275301161278372685052123, −11.19066304266943172941681183616, −10.23800956304742450762423729161, −9.197584497428101265177937416231, −8.055559255408290267204868198912, −6.83739125110649669950168179093, −6.23813642288344177179792035487, −4.96730409288645003091068534247, −4.01521440514018149047565493523, −2.20193480245002861015690810547, 0.15318965919525443180489395040, 2.26396856518051064239141814979, 4.28297207368549608927688091242, 5.02321604203734350278185867063, 6.06020796062949338468794625656, 7.27836504222611163245754026877, 7.955032282760399564812682060679, 9.352449216747396124019523645189, 10.42081484073809222789373286843, 10.80395138380351188284516310083

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