↑

Properties

Label 2-360-360.293-c1-0-62
Degree $2$
Conductor $360$
Sign $-0.0572 + 0.998i$
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  + (0.366 + 1.36i)2-s + (0.866 − 1.5i)3-s + (−1.73 + i)4-s + (−1.23 − 1.86i)5-s + (2.36 + 0.633i)6-s + (−4.23 − 1.13i)7-s + (−2 − 1.99i)8-s + (−1.5 − 2.59i)9-s + (2.09 − 2.36i)10-s + (1 + 1.73i)11-s + 3.46i·12-s + (−0.464 − 1.73i)13-s − 6.19i·14-s + (−3.86 + 0.232i)15-s + (1.99 − 3.46i)16-s + (−3 − 3i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (0.499 − 0.866i)3-s + (−0.866 + 0.5i)4-s + (−0.550 − 0.834i)5-s + (0.965 + 0.258i)6-s + (−1.59 − 0.428i)7-s + (−0.707 − 0.707i)8-s + (−0.5 − 0.866i)9-s + (0.663 − 0.748i)10-s + (0.301 + 0.522i)11-s + 0.999i·12-s + (−0.128 − 0.480i)13-s − 1.65i·14-s + (−0.998 + 0.0599i)15-s + (0.499 − 0.866i)16-s + (−0.727 − 0.727i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0572 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0572 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.0572 + 0.998i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.0572 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.570504 - 0.604162i\)
\(L(\frac12)\) \(\approx\) \(0.570504 - 0.604162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
3 \( 1 + (-0.866 + 1.5i)T \)
5 \( 1 + (1.23 + 1.86i)T \)
good7 \( 1 + (4.23 + 1.13i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.464 + 1.73i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (3 + 3i)T + 17iT^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 + (-0.232 - 0.866i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.76 - 1.59i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.464 - 0.464i)T + 37iT^{2} \)
41 \( 1 + (5.59 + 3.23i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.83 - 2.36i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.401 - 1.5i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.19 + 5.19i)T + 53iT^{2} \)
59 \( 1 + (1.56 + 0.901i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.6 + 7.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.33 + 1.96i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + (10.1 + 10.1i)T + 73iT^{2} \)
79 \( 1 + (1.90 - 1.09i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.03 + 11.3i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 2.66T + 89T^{2} \)
97 \( 1 + (6.83 + 1.83i)T + (84.0 + 48.5i)T^{2} \)
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

−11.67569114648395112179669087949, −9.613278858569849121137009349625, −9.323153087849585912073367116585, −8.137029556860488347305313635560, −7.30644559387283716890224394293, −6.66922759777581424519476019324, −5.52293360976887690021326651482, −4.10959225765517939633917953384, −3.09706688560279812727649510890, −0.48351030286172244255775552869, 2.63599386515490436787390014667, 3.37385539124051500239328264006, 4.18092284856304475780353251093, 5.69062507840100974981861511378, 6.76617208939441047528234753472, 8.376822729457030897708478590697, 9.238599377080067973025957492732, 9.928621739425485214906320507193, 10.71673787614588578242454886321, 11.54249781573277287920574794022

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