Properties

Label 2-2e6-4.3-c8-0-5
Degree $2$
Conductor $64$
Sign $-1$
Analytic cond. $26.0722$
Root an. cond. $5.10609$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 141. i·3-s + 510·5-s + 2.55e3i·7-s − 1.35e4·9-s + 1.91e4i·11-s + 2.77e4·13-s + 7.24e4i·15-s + 5.03e4·17-s + 1.08e5i·19-s − 3.62e5·21-s − 1.76e5i·23-s − 1.30e5·25-s − 9.99e5i·27-s − 5.49e4·29-s − 1.17e6i·31-s + ⋯
L(s)  = 1  + 1.75i·3-s + 0.816·5-s + 1.06i·7-s − 2.07·9-s + 1.30i·11-s + 0.970·13-s + 1.43i·15-s + 0.603·17-s + 0.833i·19-s − 1.86·21-s − 0.630i·23-s − 0.334·25-s − 1.88i·27-s − 0.0777·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(26.0722\)
Root analytic conductor: \(5.10609\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :4),\ -1)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.99907i\)
\(L(\frac12)\) \(\approx\) \(1.99907i\)
\(L(5)\) 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 \( 1 - 141. iT - 6.56e3T^{2} \)
5 \( 1 - 510T + 3.90e5T^{2} \)
7 \( 1 - 2.55e3iT - 5.76e6T^{2} \)
11 \( 1 - 1.91e4iT - 2.14e8T^{2} \)
13 \( 1 - 2.77e4T + 8.15e8T^{2} \)
17 \( 1 - 5.03e4T + 6.97e9T^{2} \)
19 \( 1 - 1.08e5iT - 1.69e10T^{2} \)
23 \( 1 + 1.76e5iT - 7.83e10T^{2} \)
29 \( 1 + 5.49e4T + 5.00e11T^{2} \)
31 \( 1 + 1.17e6iT - 8.52e11T^{2} \)
37 \( 1 + 7.93e5T + 3.51e12T^{2} \)
41 \( 1 + 7.55e4T + 7.98e12T^{2} \)
43 \( 1 - 4.99e5iT - 1.16e13T^{2} \)
47 \( 1 + 2.86e6iT - 2.38e13T^{2} \)
53 \( 1 + 1.11e7T + 6.22e13T^{2} \)
59 \( 1 + 2.18e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.38e7T + 1.91e14T^{2} \)
67 \( 1 - 7.49e6iT - 4.06e14T^{2} \)
71 \( 1 - 1.00e7iT - 6.45e14T^{2} \)
73 \( 1 - 6.51e6T + 8.06e14T^{2} \)
79 \( 1 - 4.87e7iT - 1.51e15T^{2} \)
83 \( 1 - 7.34e7iT - 2.25e15T^{2} \)
89 \( 1 - 8.67e7T + 3.93e15T^{2} \)
97 \( 1 + 4.66e7T + 7.83e15T^{2} \)
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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

−14.16201726287854485747282368923, −12.57486145401775053057546387453, −11.30574302675563056006778267263, −10.02064722060056510810802516437, −9.560503398474940627893706430049, −8.386105407345175012495295137309, −6.04969144757121573360728882367, −5.12723086185503322900284643538, −3.77928385935194435663005812495, −2.18318653934294933111889175088, 0.67580426275643655029046169741, 1.54187561082246158699524994837, 3.22086672365923328395314936354, 5.69903920157139165754757256984, 6.61963186998158943718667266761, 7.72374643027032635671405508560, 8.865468779469287258832605893093, 10.62511150733703880807589528210, 11.65670796818522996740271549094, 13.05945318276671579589730156089

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