Properties

Label 2-6e4-9.7-c1-0-8
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  + (−1.5 − 2.59i)5-s + (−2 + 3.46i)7-s + (0.5 + 0.866i)13-s + 3·17-s + 4·19-s + (−2 + 3.46i)25-s + (4.5 − 7.79i)29-s + (−2 − 3.46i)31-s + 12·35-s − 37-s + (3 + 5.19i)41-s + (4 − 6.92i)43-s + (6 − 10.3i)47-s + (−4.49 − 7.79i)49-s + 6·53-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + (−0.755 + 1.30i)7-s + (0.138 + 0.240i)13-s + 0.727·17-s + 0.917·19-s + (−0.400 + 0.692i)25-s + (0.835 − 1.44i)29-s + (−0.359 − 0.622i)31-s + 2.02·35-s − 0.164·37-s + (0.468 + 0.811i)41-s + (0.609 − 1.05i)43-s + (0.875 − 1.51i)47-s + (−0.642 − 1.11i)49-s + 0.824·53-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

\(L(1)\) \(\approx\) \(1.268855031\)
\(L(\frac12)\) \(\approx\) \(1.268855031\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4 + 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.425186132969605128752203026029, −8.766467431775290903638156691973, −8.137259291917816043118933199717, −7.22031512434760449912064748604, −5.99649696803467346375902529447, −5.45783566444143819342691536576, −4.44449747710178832331294550631, −3.45603402797998639510096728836, −2.30852222462586609344772542945, −0.70479586560519622839123601216, 0.995591272542316057194106228394, 3.00039605802937088640114894477, 3.42929634736446864710308860066, 4.39709510210782604752216641986, 5.69100725606541307670848786556, 6.72467639090079128148832262360, 7.28185181323512946689732896319, 7.77398447808587926281510681957, 9.013913198769548838075093542767, 9.971464619570892566025350264745

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