↑

Properties

Label 2-864-9.2-c2-0-2
Degree $2$
Conductor $864$
Sign $-0.780 - 0.625i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (6.30 − 3.63i)5-s + (−4.46 + 7.74i)7-s + (−10.5 − 6.08i)11-s + (−1.73 − 3.00i)13-s + 33.3i·17-s − 33.3·19-s + (−15.8 + 9.16i)23-s + (13.9 − 24.2i)25-s + (−13.6 − 7.85i)29-s + (6.66 + 11.5i)31-s + 65.0i·35-s − 35.3·37-s + (−4.55 + 2.62i)41-s + (20.1 − 34.9i)43-s + (29.9 + 17.2i)47-s + ⋯
L(s)  = 1  + (1.26 − 0.727i)5-s + (−0.638 + 1.10i)7-s + (−0.958 − 0.553i)11-s + (−0.133 − 0.231i)13-s + 1.95i·17-s − 1.75·19-s + (−0.690 + 0.398i)23-s + (0.558 − 0.968i)25-s + (−0.469 − 0.270i)29-s + (0.215 + 0.372i)31-s + 1.85i·35-s − 0.956·37-s + (−0.111 + 0.0640i)41-s + (0.469 − 0.812i)43-s + (0.637 + 0.367i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6606887612\)
\(L(\frac12)\) \(\approx\) \(0.6606887612\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-6.30 + 3.63i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (4.46 - 7.74i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.5 + 6.08i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (1.73 + 3.00i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 33.3iT - 289T^{2} \)
19 \( 1 + 33.3T + 361T^{2} \)
23 \( 1 + (15.8 - 9.16i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (13.6 + 7.85i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-6.66 - 11.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 35.3T + 1.36e3T^{2} \)
41 \( 1 + (4.55 - 2.62i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-20.1 + 34.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-29.9 - 17.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 15.9iT - 2.80e3T^{2} \)
59 \( 1 + (16.8 - 9.70i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.12 + 12.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (16.8 + 29.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 21.5iT - 5.04e3T^{2} \)
73 \( 1 - 35.0T + 5.32e3T^{2} \)
79 \( 1 + (75.5 - 130. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-107. - 62.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 58.9iT - 7.92e3T^{2} \)
97 \( 1 + (43.2 - 74.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−10.31117237018221523209813143524, −9.384832201276182837204016794665, −8.643939636870364896450484333666, −8.125988346037981950058721816403, −6.47755416343697063047679761894, −5.83243667668603924916652936961, −5.37783260039226569598165938187, −3.97731428877827932983276341526, −2.55843662476502723451462132694, −1.76862002841907408786818491462, 0.19123558052287117443161918066, 2.07218503957746347984291045877, 2.86204218801214152385674997793, 4.22283146370193264669031610602, 5.24555622566095279425613842563, 6.35490399474953389957909821683, 6.93535374359344082217309029254, 7.69717631048480851743682868550, 9.057280178929481940571223749055, 9.855177543568564516837001011204

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