↑

Properties

Label 2-6e4-9.7-c1-0-11
Degree $2$
Conductor $1296$
Sign $0.766 + 0.642i$
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  + (−0.5 + 0.866i)7-s + (−2.5 − 4.33i)13-s + 7·19-s + (2.5 − 4.33i)25-s + (−2 − 3.46i)31-s + 11·37-s + (4 − 6.92i)43-s + (3 + 5.19i)49-s + (0.5 − 0.866i)61-s + (2.5 + 4.33i)67-s − 7·73-s + (8.5 − 14.7i)79-s + 5·91-s + (9.5 − 16.4i)97-s + (−6.5 − 11.2i)103-s + ⋯
L(s)  = 1  + (−0.188 + 0.327i)7-s + (−0.693 − 1.20i)13-s + 1.60·19-s + (0.5 − 0.866i)25-s + (−0.359 − 0.622i)31-s + 1.80·37-s + (0.609 − 1.05i)43-s + (0.428 + 0.742i)49-s + (0.0640 − 0.110i)61-s + (0.305 + 0.529i)67-s − 0.819·73-s + (0.956 − 1.65i)79-s + 0.524·91-s + (0.964 − 1.67i)97-s + (−0.640 − 1.10i)103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.529954037\)
\(L(\frac12)\) \(\approx\) \(1.529954037\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
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 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 11T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
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 \( 1 + (-2.5 - 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-8.5 + 14.7i)T + (-39.5 - 68.4i)T^{2} \)
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

−9.637954757352023195353773225988, −8.812043376342046742750524825566, −7.79759446672018945723047628869, −7.32360233866093777830243244572, −6.08930148838864866561545969630, −5.44747665364545683146625030746, −4.49588486129804469509521990774, −3.24947773180374233624660994484, −2.45071826489332977874040543854, −0.74946665045970302009324785232, 1.19138592078966392478662435604, 2.57839106648213412992233862389, 3.65244167524193960820034231533, 4.65174669802628260665949362265, 5.48384101687474692311236093692, 6.59953042449922657315845705552, 7.26743801739296391238585073693, 7.990532556592202806296267877566, 9.323182531683686274546260719397, 9.436609579693224932524319866018

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