↑

Properties

Label 2-360-120.77-c1-0-1
Degree $2$
Conductor $360$
Sign $-0.996 - 0.0859i$
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  + (0.473 + 1.33i)2-s + (−1.55 + 1.26i)4-s + (−1.45 + 1.69i)5-s + (1.53 + 1.53i)7-s + (−2.41 − 1.47i)8-s + (−2.95 − 1.13i)10-s − 2.72·11-s + (0.857 + 0.857i)13-s + (−1.31 + 2.76i)14-s + (0.818 − 3.91i)16-s + (−2.55 + 2.55i)17-s − 3.54·19-s + (0.113 − 4.47i)20-s + (−1.28 − 3.63i)22-s + (0.626 + 0.626i)23-s + ⋯
L(s)  = 1  + (0.334 + 0.942i)2-s + (−0.776 + 0.630i)4-s + (−0.650 + 0.759i)5-s + (0.578 + 0.578i)7-s + (−0.853 − 0.520i)8-s + (−0.933 − 0.358i)10-s − 0.821·11-s + (0.237 + 0.237i)13-s + (−0.351 + 0.738i)14-s + (0.204 − 0.978i)16-s + (−0.619 + 0.619i)17-s − 0.812·19-s + (0.0254 − 0.999i)20-s + (−0.274 − 0.774i)22-s + (0.130 + 0.130i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $-0.996 - 0.0859i$
Analytic conductor: \(2.87461\)
Root analytic conductor: \(1.69546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :1/2),\ -0.996 - 0.0859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0452732 + 1.05199i\)
\(L(\frac12)\) \(\approx\) \(0.0452732 + 1.05199i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.473 - 1.33i)T \)
3 \( 1 \)
5 \( 1 + (1.45 - 1.69i)T \)
good7 \( 1 + (-1.53 - 1.53i)T + 7iT^{2} \)
11 \( 1 + 2.72T + 11T^{2} \)
13 \( 1 + (-0.857 - 0.857i)T + 13iT^{2} \)
17 \( 1 + (2.55 - 2.55i)T - 17iT^{2} \)
19 \( 1 + 3.54T + 19T^{2} \)
23 \( 1 + (-0.626 - 0.626i)T + 23iT^{2} \)
29 \( 1 - 5.12iT - 29T^{2} \)
31 \( 1 - 7.89T + 31T^{2} \)
37 \( 1 + (4.21 - 4.21i)T - 37iT^{2} \)
41 \( 1 - 12.4iT - 41T^{2} \)
43 \( 1 + (-5.67 - 5.67i)T + 43iT^{2} \)
47 \( 1 + (-9.45 + 9.45i)T - 47iT^{2} \)
53 \( 1 + (-6.46 + 6.46i)T - 53iT^{2} \)
59 \( 1 + 2.51iT - 59T^{2} \)
61 \( 1 - 9.49iT - 61T^{2} \)
67 \( 1 + (-9.91 + 9.91i)T - 67iT^{2} \)
71 \( 1 - 2.19iT - 71T^{2} \)
73 \( 1 + (5.71 - 5.71i)T - 73iT^{2} \)
79 \( 1 - 12.7iT - 79T^{2} \)
83 \( 1 + (-3.58 + 3.58i)T - 83iT^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (1.29 + 1.29i)T + 97iT^{2} \)
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

−11.94731331522083401498806206714, −11.07607855514927626667993571391, −10.05888152484008170758544338517, −8.564062500806996381891952210622, −8.216052083806275155635682115192, −7.05973777304236246936082959976, −6.25259811717467523716362375024, −5.06350912343492496567223034246, −4.04514040201568248033954083635, −2.69025150632413874883008017189, 0.65415185008106761431502889709, 2.40557531816273403855178576348, 3.99704927776410382214747599401, 4.67222081803337498005953191382, 5.73384837709926530169963829123, 7.40522495945251875646634465096, 8.393701281735519136892332300483, 9.155648455073644510278462155889, 10.46518182855039877271921374800, 10.95156017553403848102799382037

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