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Properties

Label 2-360-9.4-c1-0-3
Degree $2$
Conductor $360$
Sign $0.946 - 0.321i$
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  + (−0.574 − 1.63i)3-s + (−0.5 + 0.866i)5-s + (2.28 + 3.96i)7-s + (−2.33 + 1.87i)9-s + (2.29 + 3.97i)11-s + (1.83 − 3.18i)13-s + (1.70 + 0.319i)15-s − 2.55·17-s + 4.76·19-s + (5.15 − 6.01i)21-s + (1.28 − 2.22i)23-s + (−0.499 − 0.866i)25-s + (4.41 + 2.74i)27-s + (−0.956 − 1.65i)29-s + (−1.73 + 3.00i)31-s + ⋯
L(s)  = 1  + (−0.331 − 0.943i)3-s + (−0.223 + 0.387i)5-s + (0.864 + 1.49i)7-s + (−0.779 + 0.625i)9-s + (0.692 + 1.19i)11-s + (0.510 − 0.883i)13-s + (0.439 + 0.0824i)15-s − 0.619·17-s + 1.09·19-s + (1.12 − 1.31i)21-s + (0.268 − 0.464i)23-s + (−0.0999 − 0.173i)25-s + (0.849 + 0.528i)27-s + (−0.177 − 0.307i)29-s + (−0.311 + 0.539i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.946 - 0.321i$
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: \((2,\ 360,\ (\ :1/2),\ 0.946 - 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24506 + 0.205569i\)
\(L(\frac12)\) \(\approx\) \(1.24506 + 0.205569i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.574 + 1.63i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (-2.28 - 3.96i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.29 - 3.97i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.83 + 3.18i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 - 4.76T + 19T^{2} \)
23 \( 1 + (-1.28 + 2.22i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.956 + 1.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.73 - 3.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.46T + 37T^{2} \)
41 \( 1 + (3.32 - 5.75i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.27 - 5.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.01 + 8.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.03T + 53T^{2} \)
59 \( 1 + (6.13 - 10.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.79 + 8.30i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.66 + 9.82i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.67T + 71T^{2} \)
73 \( 1 - 3.12T + 73T^{2} \)
79 \( 1 + (-6.64 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.39 + 9.34i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.04T + 89T^{2} \)
97 \( 1 + (8.68 + 15.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.63403215234694085120777930902, −10.93364365437228489797697275313, −9.535366020808491234824299797541, −8.502041056597958014916531248543, −7.74064615645234475675989746902, −6.71228879635825961244449304257, −5.73891066027984429078563448489, −4.77732474060993496289900648828, −2.84821550670487985983912884597, −1.66988218802515778287475990314, 1.04599996729984834971991063634, 3.62195394564928991158855306467, 4.23492622848797213925128465380, 5.30934845031732719817677456112, 6.53608967693021439739275093761, 7.70606434316791612760401279425, 8.788371442062049383901520309812, 9.503793511462040034569952901120, 10.77061286050221355869134809223, 11.21625857266905205640271720653

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