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Properties

Label 2-6e4-9.2-c2-0-38
Degree $2$
Conductor $1296$
Sign $-0.996 + 0.0871i$
Analytic cond. $35.3134$
Root an. cond. $5.94251$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.67 + 0.965i)5-s + (2.36 − 4.09i)7-s + (−8.81 − 5.08i)11-s + (0.767 + 1.33i)13-s − 12.7i·17-s + 19.1·19-s + (−0.0879 + 0.0507i)23-s + (−10.6 + 18.4i)25-s + (−0.208 − 0.120i)29-s + (11.0 + 19.1i)31-s + 9.14i·35-s − 71.6·37-s + (−65.4 + 37.7i)41-s + (32.7 − 56.7i)43-s + (−45.1 − 26.0i)47-s + ⋯
L(s)  = 1  + (−0.334 + 0.193i)5-s + (0.338 − 0.585i)7-s + (−0.801 − 0.462i)11-s + (0.0590 + 0.102i)13-s − 0.750i·17-s + 1.00·19-s + (−0.00382 + 0.00220i)23-s + (−0.425 + 0.736i)25-s + (−0.00717 − 0.00414i)29-s + (0.356 + 0.617i)31-s + 0.261i·35-s − 1.93·37-s + (−1.59 + 0.921i)41-s + (0.761 − 1.31i)43-s + (−0.959 − 0.554i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.996 + 0.0871i$
Analytic conductor: \(35.3134\)
Root analytic conductor: \(5.94251\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ -0.996 + 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2771097362\)
\(L(\frac12)\) \(\approx\) \(0.2771097362\)
\(L(2)\) 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 \( 1 + (1.67 - 0.965i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-2.36 + 4.09i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (8.81 + 5.08i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.767 - 1.33i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.7iT - 289T^{2} \)
19 \( 1 - 19.1T + 361T^{2} \)
23 \( 1 + (0.0879 - 0.0507i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (0.208 + 0.120i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-11.0 - 19.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 71.6T + 1.36e3T^{2} \)
41 \( 1 + (65.4 - 37.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-32.7 + 56.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (45.1 + 26.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 38.6iT - 2.80e3T^{2} \)
59 \( 1 + (6.14 - 3.54i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (5.27 - 9.14i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.34 - 9.26i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 98.6iT - 5.04e3T^{2} \)
73 \( 1 + 85.9T + 5.32e3T^{2} \)
79 \( 1 + (-60.0 + 103. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (126. + 73.0i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 27.6iT - 7.92e3T^{2} \)
97 \( 1 + (-3.05 + 5.28i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.994147871771316005464606097610, −8.201397911996859840447484068978, −7.38814264201540469569139274804, −6.84830933007743467464055906689, −5.51351546569366749744050196609, −4.93210941738838743005633849720, −3.70639755676632676624191549741, −2.94195309624630801306209593360, −1.47500096306515445098149711764, −0.079011762331868145833929197359, 1.58350913882053926774992500680, 2.70204996737682937124702897525, 3.82217049059385990206816616484, 4.89365836302046799553078915640, 5.54513917923045273073655512434, 6.56242259418518274569835198005, 7.61907796219403664788767363245, 8.176490264317348023385937372508, 8.952501186991347351351326029864, 9.923891254060608868796055760679

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