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Properties

Label 2-6e4-9.2-c2-0-14
Degree $2$
Conductor $1296$
Sign $0.642 - 0.766i$
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  + (−4.89 + 2.82i)5-s + (1.5 − 2.59i)7-s + (−4.89 − 2.82i)11-s + (8.5 + 14.7i)13-s − 28.2i·17-s − 11·19-s + (34.2 − 19.7i)23-s + (3.49 − 6.06i)25-s + (−29.3 − 16.9i)29-s + (25 + 43.3i)31-s + 16.9i·35-s − 33·37-s + (−29.3 + 16.9i)41-s + (5 − 8.66i)43-s + (73.4 + 42.4i)47-s + ⋯
L(s)  = 1  + (−0.979 + 0.565i)5-s + (0.214 − 0.371i)7-s + (−0.445 − 0.257i)11-s + (0.653 + 1.13i)13-s − 1.66i·17-s − 0.578·19-s + (1.49 − 0.860i)23-s + (0.139 − 0.242i)25-s + (−1.01 − 0.585i)29-s + (0.806 + 1.39i)31-s + 0.484i·35-s − 0.891·37-s + (−0.716 + 0.413i)41-s + (0.116 − 0.201i)43-s + (1.56 + 0.902i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(35.3134\)
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: \((2,\ 1296,\ (\ :1),\ 0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.368100176\)
\(L(\frac12)\) \(\approx\) \(1.368100176\)
\(L(2)\) 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 \( 1 + (4.89 - 2.82i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.89 + 2.82i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.5 - 14.7i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 28.2iT - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
23 \( 1 + (-34.2 + 19.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (29.3 + 16.9i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-25 - 43.3i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 33T + 1.36e3T^{2} \)
41 \( 1 + (29.3 - 16.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-73.4 - 42.4i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 11.3iT - 2.80e3T^{2} \)
59 \( 1 + (24.4 - 14.1i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-20.5 + 35.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-41.5 - 71.8i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 22.6iT - 5.04e3T^{2} \)
73 \( 1 - 127T + 5.32e3T^{2} \)
79 \( 1 + (-9.5 + 16.4i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-107. - 62.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 84.8iT - 7.92e3T^{2} \)
97 \( 1 + (83.5 - 144. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.440423777645045228228259847106, −8.765599358594176484823685972165, −7.87456159848416658139906215457, −7.05463252906143742833772718375, −6.61391876036426786144006287422, −5.20530504622256874051400203919, −4.38831378027628942185028226903, −3.47483429540856317523389963956, −2.51018752575288961567318869472, −0.873567905221876954757267982984, 0.54343932436196388978073348990, 1.92134590302240437343837800140, 3.36836370806873358183697973886, 4.07499466928882617590053310518, 5.18701614925823394195770655072, 5.83292315469408645343065882535, 7.03197743846962860341741142518, 7.985153824736990318520372944338, 8.372753472235891070374922964435, 9.135438766906181745330022786070

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