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Properties

Label 2-6e4-9.4-c1-0-6
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

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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: $\chi_{1296} (433, \cdot )$
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 \( 1 + (-9.5 - 16.4i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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