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Properties

Label 2-6e4-36.7-c2-0-38
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  + (−0.866 − 1.5i)5-s + (9 + 5.19i)7-s + (−15.5 − 9i)11-s + (2.5 + 4.33i)13-s − 12.1·17-s − 10.3i·19-s + (−15.5 + 9i)23-s + (11 − 19.0i)25-s + (11.2 − 19.5i)29-s + (−18 + 10.3i)31-s − 18i·35-s − 19·37-s + (−31.1 − 54i)41-s + (−45 − 25.9i)43-s + (62.3 + 36i)47-s + ⋯
L(s)  = 1  + (−0.173 − 0.300i)5-s + (1.28 + 0.742i)7-s + (−1.41 − 0.818i)11-s + (0.192 + 0.333i)13-s − 0.713·17-s − 0.546i·19-s + (−0.677 + 0.391i)23-s + (0.440 − 0.762i)25-s + (0.388 − 0.672i)29-s + (−0.580 + 0.335i)31-s − 0.514i·35-s − 0.513·37-s + (−0.760 − 1.31i)41-s + (−1.04 − 0.604i)43-s + (1.32 + 0.765i)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: $\chi_{1296} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8715192315\)
\(L(\frac12)\) \(\approx\) \(0.8715192315\)
\(L(2)\) 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 \( 1 + (0.866 + 1.5i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-9 - 5.19i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (15.5 + 9i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.1T + 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + (15.5 - 9i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.2 + 19.5i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (18 - 10.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 19T + 1.36e3T^{2} \)
41 \( 1 + (31.1 + 54i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (45 + 25.9i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-62.3 - 36i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 62.3T + 2.80e3T^{2} \)
59 \( 1 + (-62.3 + 36i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-21.5 + 37.2i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-81 + 46.7i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 90iT - 5.04e3T^{2} \)
73 \( 1 + 107T + 5.32e3T^{2} \)
79 \( 1 + (-63 - 36.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (31.1 + 18i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 112.T + 7.92e3T^{2} \)
97 \( 1 + (-43 + 74.4i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−8.813694226054821107367266576591, −8.459282247943047343890448468458, −7.79317314768490771184541564883, −6.70947509402173199369857337056, −5.53762927548376280575023921305, −5.09326927863874247926826226699, −4.09883741200350406446827526110, −2.72585643939664066648573405092, −1.84170101328515043989131824902, −0.24097673231710291588989224961, 1.43430706612739179997268705015, 2.50008279025537360147821718083, 3.77049987162676625176385558565, 4.76894939143157439927372166855, 5.31136412616635254251239599214, 6.63213290033981990078812838549, 7.49853372218103322939669067491, 7.972864239782656503530787278318, 8.759062444257260251153881901491, 10.07047642397394405449660906330

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