↑

Properties

Label 2-360-45.4-c1-0-13
Degree $2$
Conductor $360$
Sign $0.595 + 0.803i$
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 + (2.23 + 0.133i)5-s + (0.866 − 0.5i)7-s + (−1.5 − 2.59i)9-s + (−1 − 1.73i)11-s + (1.73 + i)13-s + (2.13 − 3.23i)15-s + 6i·17-s − 2·19-s − 1.73i·21-s + (−0.866 − 0.5i)23-s + (4.96 + 0.598i)25-s − 5.19·27-s + (−3.5 − 6.06i)29-s + (3 − 5.19i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.866i)3-s + (0.998 + 0.0599i)5-s + (0.327 − 0.188i)7-s + (−0.5 − 0.866i)9-s + (−0.301 − 0.522i)11-s + (0.480 + 0.277i)13-s + (0.550 − 0.834i)15-s + 1.45i·17-s − 0.458·19-s − 0.377i·21-s + (−0.180 − 0.104i)23-s + (0.992 + 0.119i)25-s − 1.00·27-s + (−0.649 − 1.12i)29-s + (0.538 − 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62352 - 0.817170i\)
\(L(\frac12)\) \(\approx\) \(1.62352 - 0.817170i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.133i)T \)
good7 \( 1 + (-0.866 + 0.5i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.73 - i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.5 + 6.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (2.5 - 4.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-10.3 + 6i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.79 - 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 8iT - 53T^{2} \)
59 \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.33 + 2.5i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (13.8 - 8i)T + (48.5 - 84.0i)T^{2} \)
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

−11.27820271060270940436476071505, −10.41154318403150057113466550168, −9.315885905252471450648864628727, −8.446076383730760544529729604479, −7.65827593749543850263835344795, −6.29697638736949186263284442152, −5.89165328110235189086084430498, −4.11628447072577090996883996598, −2.62097249530466873784156637638, −1.46047502354722191220870568743, 2.05538764258521715343308577413, 3.25605554970502985338184883907, 4.80820835615152648658151524934, 5.40818015185060944672442689470, 6.79150902713689313699145238147, 8.060093109220541383188053156963, 9.017372079178075469012277439002, 9.675276273608625394229351425580, 10.51933864933667146631655878487, 11.30518943135794251724939319703

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