Properties

Label 2-2e6-64.5-c1-0-6
Degree $2$
Conductor $64$
Sign $-0.932 + 0.361i$
Analytic cond. $0.511042$
Root an. cond. $0.714872$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.599 − 1.28i)2-s + (−2.03 − 1.36i)3-s + (−1.28 + 1.53i)4-s + (−1.53 − 0.306i)5-s + (−0.521 + 3.42i)6-s + (1.01 − 2.44i)7-s + (2.73 + 0.718i)8-s + (1.15 + 2.78i)9-s + (0.530 + 2.15i)10-s + (−2.71 − 4.06i)11-s + (4.70 − 1.38i)12-s + (4.23 − 0.841i)13-s + (−3.73 + 0.169i)14-s + (2.72 + 2.72i)15-s + (−0.720 − 3.93i)16-s + (0.228 − 0.228i)17-s + ⋯
L(s)  = 1  + (−0.424 − 0.905i)2-s + (−1.17 − 0.786i)3-s + (−0.640 + 0.768i)4-s + (−0.688 − 0.136i)5-s + (−0.213 + 1.39i)6-s + (0.382 − 0.924i)7-s + (0.967 + 0.253i)8-s + (0.384 + 0.928i)9-s + (0.167 + 0.681i)10-s + (−0.818 − 1.22i)11-s + (1.35 − 0.400i)12-s + (1.17 − 0.233i)13-s + (−0.999 + 0.0453i)14-s + (0.702 + 0.702i)15-s + (−0.180 − 0.983i)16-s + (0.0553 − 0.0553i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.932 + 0.361i$
Analytic conductor: \(0.511042\)
Root analytic conductor: \(0.714872\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :1/2),\ -0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0776134 - 0.414708i\)
\(L(\frac12)\) \(\approx\) \(0.0776134 - 0.414708i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.599 + 1.28i)T \)
good3 \( 1 + (2.03 + 1.36i)T + (1.14 + 2.77i)T^{2} \)
5 \( 1 + (1.53 + 0.306i)T + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (-1.01 + 2.44i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (2.71 + 4.06i)T + (-4.20 + 10.1i)T^{2} \)
13 \( 1 + (-4.23 + 0.841i)T + (12.0 - 4.97i)T^{2} \)
17 \( 1 + (-0.228 + 0.228i)T - 17iT^{2} \)
19 \( 1 + (-1.30 - 6.57i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-2.81 + 1.16i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.67 + 2.50i)T + (-11.0 - 26.7i)T^{2} \)
31 \( 1 - 1.06iT - 31T^{2} \)
37 \( 1 + (-2.13 + 10.7i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (2.57 - 1.06i)T + (28.9 - 28.9i)T^{2} \)
43 \( 1 + (-0.575 + 0.384i)T + (16.4 - 39.7i)T^{2} \)
47 \( 1 + (-2.61 + 2.61i)T - 47iT^{2} \)
53 \( 1 + (-2.60 - 3.89i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (-8.66 - 1.72i)T + (54.5 + 22.5i)T^{2} \)
61 \( 1 + (-6.23 - 4.16i)T + (23.3 + 56.3i)T^{2} \)
67 \( 1 + (6.91 + 4.62i)T + (25.6 + 61.8i)T^{2} \)
71 \( 1 + (0.606 - 1.46i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (1.36 + 3.30i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-4.69 - 4.69i)T + 79iT^{2} \)
83 \( 1 + (-0.803 - 4.04i)T + (-76.6 + 31.7i)T^{2} \)
89 \( 1 + (10.3 + 4.28i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + 7.30iT - 97T^{2} \)
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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

−13.90978109876032502453878545305, −13.03971543551922185577614923264, −11.97862608344340626246169070128, −11.10231420327250543441767509499, −10.50103430132412654925582276879, −8.390753505548714309521949782394, −7.50831305534853910380043177124, −5.70104272112595851929131148444, −3.82218162177305325302885545505, −0.851016285174800219392471059842, 4.58985882046125990089228946280, 5.49099840532314304828046791238, 6.93053379476970433205584586993, 8.391522676460491417324166994548, 9.692807624383801136562279388417, 10.89314794068330323643627952323, 11.73707055508039699737168418414, 13.29445197639948162643674828255, 15.15993847657766074320311293593, 15.46000472165879934460045740482

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