↑

Properties

Label 2-3-3.2-c64-0-14
Degree $2$
Conductor $3$
Sign $0.977 - 0.212i$
Analytic cond. $77.8210$
Root an. cond. $8.82162$
Motivic weight $64$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3.00e9i·2-s + (1.81e15 − 3.93e14i)3-s + 9.42e18·4-s − 2.20e22i·5-s + (1.18e24 + 5.43e24i)6-s + 9.43e26·7-s + 8.37e28i·8-s + (3.12e30 − 1.42e30i)9-s + 6.61e31·10-s − 1.69e33i·11-s + (1.70e34 − 3.71e33i)12-s + 2.86e35·13-s + 2.83e36i·14-s + (−8.67e36 − 3.98e37i)15-s − 7.76e37·16-s + 3.89e39i·17-s + ⋯
L(s)  = 1  + 0.699i·2-s + (0.977 − 0.212i)3-s + 0.510·4-s − 0.945i·5-s + (0.148 + 0.683i)6-s + 0.854·7-s + 1.05i·8-s + (0.909 − 0.415i)9-s + 0.661·10-s − 0.803i·11-s + (0.499 − 0.108i)12-s + 0.646·13-s + 0.597i·14-s + (−0.200 − 0.923i)15-s − 0.228·16-s + 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(65-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+32) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.977 - 0.212i$
Analytic conductor: \(77.8210\)
Root analytic conductor: \(8.82162\)
Motivic weight: \(64\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :32),\ 0.977 - 0.212i)\)

Particular Values

\(L(\frac{65}{2})\) \(\approx\) \(4.797621225\)
\(L(\frac12)\) \(\approx\) \(4.797621225\)
\(L(33)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.81e15 + 3.93e14i)T \)
good2 \( 1 - 3.00e9iT - 1.84e19T^{2} \)
5 \( 1 + 2.20e22iT - 5.42e44T^{2} \)
7 \( 1 - 9.43e26T + 1.21e54T^{2} \)
11 \( 1 + 1.69e33iT - 4.45e66T^{2} \)
13 \( 1 - 2.86e35T + 1.96e71T^{2} \)
17 \( 1 - 3.89e39iT - 5.60e78T^{2} \)
19 \( 1 + 1.08e40T + 6.92e81T^{2} \)
23 \( 1 + 1.19e43iT - 1.41e87T^{2} \)
29 \( 1 - 6.29e46iT - 3.92e93T^{2} \)
31 \( 1 - 6.35e47T + 2.79e95T^{2} \)
37 \( 1 + 1.98e50T + 2.31e100T^{2} \)
41 \( 1 + 6.64e51iT - 1.65e103T^{2} \)
43 \( 1 - 2.64e51T + 3.48e104T^{2} \)
47 \( 1 + 3.26e53iT - 1.03e107T^{2} \)
53 \( 1 + 1.07e55iT - 2.25e110T^{2} \)
59 \( 1 + 5.16e56iT - 2.16e113T^{2} \)
61 \( 1 - 1.92e57T + 1.82e114T^{2} \)
67 \( 1 + 2.61e58T + 7.39e116T^{2} \)
71 \( 1 - 1.40e59iT - 3.02e118T^{2} \)
73 \( 1 - 5.03e59T + 1.78e119T^{2} \)
79 \( 1 - 7.40e60T + 2.80e121T^{2} \)
83 \( 1 - 4.72e61iT - 6.62e122T^{2} \)
89 \( 1 + 5.97e61iT - 5.76e124T^{2} \)
97 \( 1 + 5.26e62T + 1.42e127T^{2} \)
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

−13.86985358702870075235382308511, −12.41441548515251986716907149645, −10.75644986687436413116007358153, −8.459334644399282491038165292914, −8.368363911694244667465248308930, −6.67642633362560135183914241626, −5.24317067170729477729110399918, −3.68522645229117558693796625650, −2.03103251902158085344645666655, −1.12359971822047049364671054999, 1.24417131886228033009548228129, 2.35454772708146827871078028877, 3.12805428130828731495313932319, 4.52070931945922518397975939030, 6.76881473055276235773944105812, 7.80407361131818211305100275605, 9.571579292241382770226560502530, 10.68706332221086897368139421084, 11.81919584864477410584420955837, 13.59429861696830091059161257421

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