Properties

Label 2-2e6-16.5-c3-0-3
Degree $2$
Conductor $64$
Sign $0.757 + 0.652i$
Analytic cond. $3.77612$
Root an. cond. $1.94322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.96 − 5.96i)3-s + (8.67 + 8.67i)5-s − 1.63i·7-s − 44.1i·9-s + (−18.2 − 18.2i)11-s + (−9.34 + 9.34i)13-s + 103.·15-s + 53.6·17-s + (−70.9 + 70.9i)19-s + (−9.77 − 9.77i)21-s + 25.1i·23-s + 25.6i·25-s + (−102. − 102. i)27-s + (−181. + 181. i)29-s − 132.·31-s + ⋯
L(s)  = 1  + (1.14 − 1.14i)3-s + (0.776 + 0.776i)5-s − 0.0885i·7-s − 1.63i·9-s + (−0.498 − 0.498i)11-s + (−0.199 + 0.199i)13-s + 1.78·15-s + 0.764·17-s + (−0.857 + 0.857i)19-s + (−0.101 − 0.101i)21-s + 0.227i·23-s + 0.205i·25-s + (−0.729 − 0.729i)27-s + (−1.15 + 1.15i)29-s − 0.768·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.757 + 0.652i$
Analytic conductor: \(3.77612\)
Root analytic conductor: \(1.94322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :3/2),\ 0.757 + 0.652i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.97913 - 0.734574i\)
\(L(\frac12)\) \(\approx\) \(1.97913 - 0.734574i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (-5.96 + 5.96i)T - 27iT^{2} \)
5 \( 1 + (-8.67 - 8.67i)T + 125iT^{2} \)
7 \( 1 + 1.63iT - 343T^{2} \)
11 \( 1 + (18.2 + 18.2i)T + 1.33e3iT^{2} \)
13 \( 1 + (9.34 - 9.34i)T - 2.19e3iT^{2} \)
17 \( 1 - 53.6T + 4.91e3T^{2} \)
19 \( 1 + (70.9 - 70.9i)T - 6.85e3iT^{2} \)
23 \( 1 - 25.1iT - 1.21e4T^{2} \)
29 \( 1 + (181. - 181. i)T - 2.43e4iT^{2} \)
31 \( 1 + 132.T + 2.97e4T^{2} \)
37 \( 1 + (-174. - 174. i)T + 5.06e4iT^{2} \)
41 \( 1 + 198. iT - 6.89e4T^{2} \)
43 \( 1 + (-285. - 285. i)T + 7.95e4iT^{2} \)
47 \( 1 + 78.3T + 1.03e5T^{2} \)
53 \( 1 + (525. + 525. i)T + 1.48e5iT^{2} \)
59 \( 1 + (46.5 + 46.5i)T + 2.05e5iT^{2} \)
61 \( 1 + (-193. + 193. i)T - 2.26e5iT^{2} \)
67 \( 1 + (282. - 282. i)T - 3.00e5iT^{2} \)
71 \( 1 + 727. iT - 3.57e5T^{2} \)
73 \( 1 + 106. iT - 3.89e5T^{2} \)
79 \( 1 - 58.9T + 4.93e5T^{2} \)
83 \( 1 + (-410. + 410. i)T - 5.71e5iT^{2} \)
89 \( 1 + 768. iT - 7.04e5T^{2} \)
97 \( 1 + 809.T + 9.12e5T^{2} \)
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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

−14.30944608249244471983468256925, −13.33296170056693285948989481345, −12.48599834639166092794286018010, −10.81432943046014674010008291681, −9.524231476470720225583141042748, −8.218057802697551169751213457741, −7.19871704857499675602393597171, −5.97074581602943873261856794782, −3.22270301478083509587652583664, −1.89007037301595103598371103169, 2.42938515056317504691844792984, 4.22737583341852780965126001110, 5.48772838860291919884581925143, 7.77714233863238067137603259673, 9.052461439264619324268312195253, 9.641703767512068370306604675726, 10.75702229455889334377378222151, 12.66339829950854306001880283379, 13.56494627844703598969120736393, 14.70125069440133760320680963535

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