↑

Properties

Label 2-360-9.7-c1-0-4
Degree $2$
Conductor $360$
Sign $0.173 - 0.984i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 1.5i)3-s + (0.5 + 0.866i)5-s + (−0.133 + 0.232i)7-s + (−1.5 + 2.59i)9-s + (0.732 − 1.26i)11-s + (2.73 + 4.73i)13-s + (−0.866 + 1.5i)15-s + 0.535·17-s − 2·19-s − 0.464·21-s + (−1.86 − 3.23i)23-s + (−0.499 + 0.866i)25-s − 5.19·27-s + (−0.767 + 1.33i)29-s + (1 + 1.73i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.223 + 0.387i)5-s + (−0.0506 + 0.0877i)7-s + (−0.5 + 0.866i)9-s + (0.220 − 0.382i)11-s + (0.757 + 1.31i)13-s + (−0.223 + 0.387i)15-s + 0.129·17-s − 0.458·19-s − 0.101·21-s + (−0.389 − 0.673i)23-s + (−0.0999 + 0.173i)25-s − 1.00·27-s + (−0.142 + 0.246i)29-s + (0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21826 + 1.02224i\)
\(L(\frac12)\) \(\approx\) \(1.21826 + 1.02224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.133 - 0.232i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.732 + 1.26i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.73 - 4.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.535T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.767 - 1.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 + (4.96 + 8.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.26 + 3.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.133 - 0.232i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (7.19 + 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.23 + 7.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.13 + 5.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 - 6.92T + 73T^{2} \)
79 \( 1 + (7.73 - 13.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.59 + 11.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.92T + 89T^{2} \)
97 \( 1 + (-4.46 + 7.73i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.31996314793636223974767493967, −10.76359502897929213909736253461, −9.727019770327727250996407362179, −8.968167763531613637408146790246, −8.176424050990081386803865532050, −6.80345614137164171079695257944, −5.82399099140721470929319998351, −4.46795034579811914521682656604, −3.55245400186546745904907568799, −2.18073709214587891063452903354, 1.15790167370062726802460172380, 2.67454307753259937510516651636, 3.99224044203238754945912941126, 5.58765064207695574089481178726, 6.42315505890095790903619972942, 7.65649043858417910371252090327, 8.277912963523894784439272872381, 9.284392011016247546117663951369, 10.20768504389424253203188056406, 11.41880067556093429114089602557

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