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Properties

Label 2-2e6-16.13-c11-0-8
Degree $2$
Conductor $64$
Sign $0.317 - 0.948i$
Analytic cond. $49.1739$
Root an. cond. $7.01241$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (435. + 435. i)3-s + (−1.51e3 + 1.51e3i)5-s − 5.97e4i·7-s + 2.01e5i·9-s + (−5.61e4 + 5.61e4i)11-s + (−3.15e5 − 3.15e5i)13-s − 1.32e6·15-s + 8.11e6·17-s + (1.33e7 + 1.33e7i)19-s + (2.59e7 − 2.59e7i)21-s + 2.84e7i·23-s + 4.42e7i·25-s + (−1.05e7 + 1.05e7i)27-s + (7.59e7 + 7.59e7i)29-s − 5.67e7·31-s + ⋯
L(s)  = 1  + (1.03 + 1.03i)3-s + (−0.217 + 0.217i)5-s − 1.34i·7-s + 1.13i·9-s + (−0.105 + 0.105i)11-s + (−0.235 − 0.235i)13-s − 0.448·15-s + 1.38·17-s + (1.23 + 1.23i)19-s + (1.38 − 1.38i)21-s + 0.922i·23-s + 0.905i·25-s + (−0.141 + 0.141i)27-s + (0.687 + 0.687i)29-s − 0.355·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.43798 + 1.75469i\)
\(L(\frac12)\) \(\approx\) \(2.43798 + 1.75469i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-435. - 435. i)T + 1.77e5iT^{2} \)
5 \( 1 + (1.51e3 - 1.51e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 5.97e4iT - 1.97e9T^{2} \)
11 \( 1 + (5.61e4 - 5.61e4i)T - 2.85e11iT^{2} \)
13 \( 1 + (3.15e5 + 3.15e5i)T + 1.79e12iT^{2} \)
17 \( 1 - 8.11e6T + 3.42e13T^{2} \)
19 \( 1 + (-1.33e7 - 1.33e7i)T + 1.16e14iT^{2} \)
23 \( 1 - 2.84e7iT - 9.52e14T^{2} \)
29 \( 1 + (-7.59e7 - 7.59e7i)T + 1.22e16iT^{2} \)
31 \( 1 + 5.67e7T + 2.54e16T^{2} \)
37 \( 1 + (-3.75e8 + 3.75e8i)T - 1.77e17iT^{2} \)
41 \( 1 - 1.51e7iT - 5.50e17T^{2} \)
43 \( 1 + (1.08e9 - 1.08e9i)T - 9.29e17iT^{2} \)
47 \( 1 + 2.79e9T + 2.47e18T^{2} \)
53 \( 1 + (-2.03e9 + 2.03e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-4.23e9 + 4.23e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-6.54e9 - 6.54e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-8.01e9 - 8.01e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 1.40e10iT - 2.31e20T^{2} \)
73 \( 1 - 6.29e9iT - 3.13e20T^{2} \)
79 \( 1 - 3.46e10T + 7.47e20T^{2} \)
83 \( 1 + (4.29e10 + 4.29e10i)T + 1.28e21iT^{2} \)
89 \( 1 - 3.39e9iT - 2.77e21T^{2} \)
97 \( 1 - 1.55e10T + 7.15e21T^{2} \)
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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

−13.09589931576452032578991375077, −11.47951338457303032733164162655, −10.09558059335347561906974654359, −9.765053034167169791876449520508, −8.112043813084914095059108417317, −7.31645030560180796805770280356, −5.24026278529309311573608983575, −3.77565881619355827061088783104, −3.24424270354043217306212941910, −1.20633680085053596186873643445, 0.809321461010590822662048984032, 2.24350533358266187362304400586, 3.08378937066438778848901733185, 5.12070617175302710896961996177, 6.60665824288089712123762643336, 7.904148886852273433902607476807, 8.631846713792875387319237863133, 9.737504035181899843392326075529, 11.74544891708036869247963407921, 12.38982488748877474978910076040

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