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Properties

Label 2-2e6-16.11-c4-0-4
Degree $2$
Conductor $64$
Sign $0.654 + 0.756i$
Analytic cond. $6.61567$
Root an. cond. $2.57209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0461 + 0.0461i)3-s + (−8.04 − 8.04i)5-s + 49.8·7-s − 80.9i·9-s + (84.2 − 84.2i)11-s + (19.4 − 19.4i)13-s − 0.743i·15-s + 437.·17-s + (−349. − 349. i)19-s + (2.30 + 2.30i)21-s + 404.·23-s − 495. i·25-s + (7.48 − 7.48i)27-s + (−1.03e3 + 1.03e3i)29-s + 1.50e3i·31-s + ⋯
L(s)  = 1  + (0.00513 + 0.00513i)3-s + (−0.321 − 0.321i)5-s + 1.01·7-s − 0.999i·9-s + (0.696 − 0.696i)11-s + (0.115 − 0.115i)13-s − 0.00330i·15-s + 1.51·17-s + (−0.966 − 0.966i)19-s + (0.00522 + 0.00522i)21-s + 0.765·23-s − 0.792i·25-s + (0.0102 − 0.0102i)27-s + (−1.22 + 1.22i)29-s + 1.56i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.654 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.654 + 0.756i$
Analytic conductor: \(6.61567\)
Root analytic conductor: \(2.57209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :2),\ 0.654 + 0.756i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.51527 - 0.692732i\)
\(L(\frac12)\) \(\approx\) \(1.51527 - 0.692732i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-0.0461 - 0.0461i)T + 81iT^{2} \)
5 \( 1 + (8.04 + 8.04i)T + 625iT^{2} \)
7 \( 1 - 49.8T + 2.40e3T^{2} \)
11 \( 1 + (-84.2 + 84.2i)T - 1.46e4iT^{2} \)
13 \( 1 + (-19.4 + 19.4i)T - 2.85e4iT^{2} \)
17 \( 1 - 437.T + 8.35e4T^{2} \)
19 \( 1 + (349. + 349. i)T + 1.30e5iT^{2} \)
23 \( 1 - 404.T + 2.79e5T^{2} \)
29 \( 1 + (1.03e3 - 1.03e3i)T - 7.07e5iT^{2} \)
31 \( 1 - 1.50e3iT - 9.23e5T^{2} \)
37 \( 1 + (434. + 434. i)T + 1.87e6iT^{2} \)
41 \( 1 - 696. iT - 2.82e6T^{2} \)
43 \( 1 + (917. - 917. i)T - 3.41e6iT^{2} \)
47 \( 1 + 111. iT - 4.87e6T^{2} \)
53 \( 1 + (-1.04e3 - 1.04e3i)T + 7.89e6iT^{2} \)
59 \( 1 + (-1.71e3 + 1.71e3i)T - 1.21e7iT^{2} \)
61 \( 1 + (-3.71e3 + 3.71e3i)T - 1.38e7iT^{2} \)
67 \( 1 + (-1.85e3 - 1.85e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 1.16e3T + 2.54e7T^{2} \)
73 \( 1 + 905. iT - 2.83e7T^{2} \)
79 \( 1 + 5.86e3iT - 3.89e7T^{2} \)
83 \( 1 + (-7.56e3 - 7.56e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 6.43e3iT - 6.27e7T^{2} \)
97 \( 1 + 413.T + 8.85e7T^{2} \)
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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

−14.31844136044951414863915108006, −12.74599483024031988574004975469, −11.79405744483561014157074279723, −10.79543150234268215441313700704, −9.157090877780694711522501846867, −8.253633191074914461500818443202, −6.72568563856566635111919731541, −5.13422894914495174597495355935, −3.55363768048893372660260028058, −1.06655273698252401344545202884, 1.83615294684446318340327511908, 4.04291265708000836430670674396, 5.51796240062243430164252348235, 7.34957371011356847017421812334, 8.206882181575434079716314559314, 9.836653457636702852865990444310, 11.04227692154258640054758046777, 11.91888657365024938656834779994, 13.28208683871634695730632400182, 14.56747451665924187253517995838

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