↑

Properties

Label 2-6e4-36.11-c1-0-10
Degree $2$
Conductor $1296$
Sign $0.642 - 0.766i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
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  + (4.5 + 2.59i)7-s + (2.5 + 4.33i)13-s − 5.19i·19-s + (−2.5 + 4.33i)25-s + (−9 + 5.19i)31-s + 11·37-s + (−9 − 5.19i)43-s + (10 + 17.3i)49-s + (0.5 − 0.866i)61-s + (13.5 − 7.79i)67-s + 7·73-s + (4.5 + 2.59i)79-s + 25.9i·91-s + (−9.5 + 16.4i)97-s + (13.5 − 7.79i)103-s + ⋯
L(s)  = 1  + (1.70 + 0.981i)7-s + (0.693 + 1.20i)13-s − 1.19i·19-s + (−0.5 + 0.866i)25-s + (−1.61 + 0.933i)31-s + 1.80·37-s + (−1.37 − 0.792i)43-s + (1.42 + 2.47i)49-s + (0.0640 − 0.110i)61-s + (1.64 − 0.952i)67-s + 0.819·73-s + (0.506 + 0.292i)79-s + 2.72i·91-s + (−0.964 + 1.67i)97-s + (1.33 − 0.767i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.013532749\)
\(L(\frac12)\) \(\approx\) \(2.013532749\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-4.5 - 2.59i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 5.19iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9 - 5.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9 + 5.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.5 + 7.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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

−9.470570090356143421798942086014, −8.940951847667308329536116191992, −8.268224373205767301190406580411, −7.40306571360408387840554494653, −6.46487655691979086427647405746, −5.38647248708788311457938252948, −4.83306592173024571555445160727, −3.77494864241125368317108564764, −2.31679278661667540193135661397, −1.50711162235389149707276890328, 0.942113657085336780854598286456, 2.03663130615716882751725528435, 3.59655871859952174450881675602, 4.33669709400533156588235149354, 5.32245723708886382928064509824, 6.09660609737121073298803901806, 7.35954374425345910510747919885, 8.046828685041279380352808564356, 8.328925532816339426799889938216, 9.718007030682469486275479393183

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