↑

Properties

Label 2-18e2-9.7-c3-0-6
Degree $2$
Conductor $324$
Sign $0.984 + 0.173i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (10.5 + 18.2i)5-s + (15.7 − 27.2i)7-s + (18.3 − 31.7i)11-s + (−28.2 − 48.9i)13-s + 35.8·17-s + 83.4·19-s + (−34.7 − 60.2i)23-s + (−159. + 276. i)25-s + (40.8 − 70.7i)29-s + (36.4 + 63.1i)31-s + 663.·35-s − 25.4·37-s + (199. + 345. i)41-s + (41.7 − 72.2i)43-s + (155. − 270i)47-s + ⋯
L(s)  = 1  + (0.942 + 1.63i)5-s + (0.850 − 1.47i)7-s + (0.502 − 0.870i)11-s + (−0.602 − 1.04i)13-s + 0.510·17-s + 1.00·19-s + (−0.315 − 0.546i)23-s + (−1.27 + 2.20i)25-s + (0.261 − 0.453i)29-s + (0.211 + 0.366i)31-s + 3.20·35-s − 0.113·37-s + (0.760 + 1.31i)41-s + (0.147 − 0.256i)43-s + (0.483 − 0.837i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.984 + 0.173i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.485339015\)
\(L(\frac12)\) \(\approx\) \(2.485339015\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10.5 - 18.2i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (-15.7 + 27.2i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-18.3 + 31.7i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (28.2 + 48.9i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 - 35.8T + 4.91e3T^{2} \)
19 \( 1 - 83.4T + 6.85e3T^{2} \)
23 \( 1 + (34.7 + 60.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-40.8 + 70.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-36.4 - 63.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 25.4T + 5.06e4T^{2} \)
41 \( 1 + (-199. - 345. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-41.7 + 72.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-155. + 270i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 4.09T + 1.48e5T^{2} \)
59 \( 1 + (-176. - 305. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (1.77 - 3.06i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (246. + 426. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 154.T + 3.57e5T^{2} \)
73 \( 1 - 305T + 3.89e5T^{2} \)
79 \( 1 + (335. - 581. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (646. - 1.12e3i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 1.18e3T + 7.04e5T^{2} \)
97 \( 1 + (307. - 533. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−10.88499174624140748387536257613, −10.38067496201083707620804931690, −9.671240130258115513631893578390, −8.012918874257326819768406153984, −7.29894973840294531670760531512, −6.37859168344086839332159213220, −5.31097573056752159496583246182, −3.73840600205194761409173401132, −2.70763001227739790006868821025, −1.05670245107132424457929804283, 1.41366243496511077079417644923, 2.21095296083330576836607996820, 4.46953687994398689298633005975, 5.21057363557827237884111799215, 5.95380348964519585757217150336, 7.53326050700629384152827588870, 8.702764449534332018982177483088, 9.255697620102469444487070840665, 9.856169312709320603231571976797, 11.63783936998944512072104049932

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