Properties

Label 2-2e6-16.5-c3-0-0
Degree $2$
Conductor $64$
Sign $-0.206 - 0.978i$
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  + (−1.98 + 1.98i)3-s + (−0.596 − 0.596i)5-s + 29.0i·7-s + 19.1i·9-s + (−12.1 − 12.1i)11-s + (−48.5 + 48.5i)13-s + 2.36·15-s + 86.7·17-s + (54.8 − 54.8i)19-s + (−57.6 − 57.6i)21-s + 70.2i·23-s − 124. i·25-s + (−91.5 − 91.5i)27-s + (63.4 − 63.4i)29-s + 8.86·31-s + ⋯
L(s)  = 1  + (−0.381 + 0.381i)3-s + (−0.0533 − 0.0533i)5-s + 1.57i·7-s + 0.708i·9-s + (−0.332 − 0.332i)11-s + (−1.03 + 1.03i)13-s + 0.0407·15-s + 1.23·17-s + (0.662 − 0.662i)19-s + (−0.599 − 0.599i)21-s + 0.636i·23-s − 0.994i·25-s + (−0.652 − 0.652i)27-s + (0.405 − 0.405i)29-s + 0.0513·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.206 - 0.978i$
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: \((2,\ 64,\ (\ :3/2),\ -0.206 - 0.978i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.671504 + 0.828081i\)
\(L(\frac12)\) \(\approx\) \(0.671504 + 0.828081i\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.98 - 1.98i)T - 27iT^{2} \)
5 \( 1 + (0.596 + 0.596i)T + 125iT^{2} \)
7 \( 1 - 29.0iT - 343T^{2} \)
11 \( 1 + (12.1 + 12.1i)T + 1.33e3iT^{2} \)
13 \( 1 + (48.5 - 48.5i)T - 2.19e3iT^{2} \)
17 \( 1 - 86.7T + 4.91e3T^{2} \)
19 \( 1 + (-54.8 + 54.8i)T - 6.85e3iT^{2} \)
23 \( 1 - 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (-63.4 + 63.4i)T - 2.43e4iT^{2} \)
31 \( 1 - 8.86T + 2.97e4T^{2} \)
37 \( 1 + (21.7 + 21.7i)T + 5.06e4iT^{2} \)
41 \( 1 - 153. iT - 6.89e4T^{2} \)
43 \( 1 + (-120. - 120. i)T + 7.95e4iT^{2} \)
47 \( 1 - 99.9T + 1.03e5T^{2} \)
53 \( 1 + (-389. - 389. i)T + 1.48e5iT^{2} \)
59 \( 1 + (-324. - 324. i)T + 2.05e5iT^{2} \)
61 \( 1 + (0.339 - 0.339i)T - 2.26e5iT^{2} \)
67 \( 1 + (565. - 565. i)T - 3.00e5iT^{2} \)
71 \( 1 + 419. iT - 3.57e5T^{2} \)
73 \( 1 + 374. iT - 3.89e5T^{2} \)
79 \( 1 - 705.T + 4.93e5T^{2} \)
83 \( 1 + (-947. + 947. i)T - 5.71e5iT^{2} \)
89 \( 1 - 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−14.85523291307456483136910499322, −13.70658305384352902670503706588, −12.18408298768799662981066093357, −11.60108896839824569354930802245, −10.11439729708169181782943475187, −9.043075354512368013186100321798, −7.67783469617989167820368840851, −5.84665860595227451021323068033, −4.84334816865138476903176497857, −2.53615521791389494222022443180, 0.789650534567216147754534854108, 3.54571937885128654035376213578, 5.33915138605416318804930165410, 7.01661750108004624773387249594, 7.77605532222556755936756068046, 9.796711792669717987335971733992, 10.57266548753053579336337603484, 12.05026963315747231437508489001, 12.86580561560083338888035775079, 14.11979687926607388830021377210

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