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Properties

Label 2-6e4-36.11-c1-0-5
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

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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

\(L(1)\) \(\approx\) \(1.162513674\)
\(L(\frac12)\) \(\approx\) \(1.162513674\)
\(L(\frac{3}{2})\) 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 \( 1 + (9.5 - 16.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.649304019542622494459145834430, −9.284618316416540551441264685151, −8.035786833222693077962625475958, −7.26871237797887970615923864674, −6.31967369512688114451687460779, −5.99850095907662745116798684150, −4.28618846900494596557595763794, −3.82291310433862024014930451539, −2.72141177884040881689139644468, −1.10906952895749879989538321646, 0.57378189885632896911946201687, 2.62462277610483802661090778208, 3.09383752512518789376061432841, 4.33629274029684484411649105487, 5.60818596065546355953762466386, 6.15784028609827297570002826187, 6.89623611599616865831638585242, 8.062739278088967632716251718838, 8.813207284656869274342659048856, 9.537419897530705178550962949928

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