↑

Properties

Label 2-360-45.4-c1-0-10
Degree $2$
Conductor $360$
Sign $0.999 - 0.00893i$
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  + (1.54 + 0.784i)3-s + (1.59 − 1.56i)5-s + (0.608 − 0.351i)7-s + (1.77 + 2.42i)9-s + (−1.80 − 3.12i)11-s + (1.97 + 1.14i)13-s + (3.69 − 1.16i)15-s + 0.471i·17-s − 5.51·19-s + (1.21 − 0.0654i)21-s + (2.68 + 1.55i)23-s + (0.113 − 4.99i)25-s + (0.834 + 5.12i)27-s + (0.195 + 0.339i)29-s + (−3.10 + 5.38i)31-s + ⋯
L(s)  = 1  + (0.891 + 0.452i)3-s + (0.715 − 0.699i)5-s + (0.230 − 0.132i)7-s + (0.590 + 0.807i)9-s + (−0.543 − 0.941i)11-s + (0.549 + 0.316i)13-s + (0.954 − 0.299i)15-s + 0.114i·17-s − 1.26·19-s + (0.265 − 0.0142i)21-s + (0.560 + 0.323i)23-s + (0.0226 − 0.999i)25-s + (0.160 + 0.987i)27-s + (0.0363 + 0.0630i)29-s + (−0.558 + 0.966i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.999 - 0.00893i$
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: \((2,\ 360,\ (\ :1/2),\ 0.999 - 0.00893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97508 + 0.00882799i\)
\(L(\frac12)\) \(\approx\) \(1.97508 + 0.00882799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.54 - 0.784i)T \)
5 \( 1 + (-1.59 + 1.56i)T \)
good7 \( 1 + (-0.608 + 0.351i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.80 + 3.12i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.97 - 1.14i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.471iT - 17T^{2} \)
19 \( 1 + 5.51T + 19T^{2} \)
23 \( 1 + (-2.68 - 1.55i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.195 - 0.339i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.10 - 5.38i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 + (-5.18 + 8.97i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.279 - 0.161i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.51 - 5.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 12.0iT - 53T^{2} \)
59 \( 1 + (-3.13 + 5.42i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.09 - 1.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.04 + 5.22i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.13T + 71T^{2} \)
73 \( 1 + 3.39iT - 73T^{2} \)
79 \( 1 + (1.66 + 2.88i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.2 - 7.06i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + (11.4 - 6.62i)T + (48.5 - 84.0i)T^{2} \)
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

−11.16353536838425703206364002509, −10.46270649272800778576472625804, −9.466544523374053697898669138854, −8.617206402158058719595325944160, −8.136422067203184072436632324289, −6.63266221870496594965139928692, −5.40957806468287204854145210048, −4.42372864221634697533229150493, −3.12988019790698685805560553402, −1.69504575621571092951995647310, 1.89725920565083635491311918410, 2.82688775356751109137892550800, 4.25896032657033944186377600523, 5.78561371622705235349130812815, 6.80958718588619856563608823262, 7.64043144368343214930722575011, 8.648559617682463382815770635126, 9.581124314612187671167784520831, 10.40015046598887227692558905890, 11.32956428762857467833378307038

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