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Properties

Label 2-2e6-16.13-c11-0-15
Degree $2$
Conductor $64$
Sign $0.911 + 0.411i$
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  + (491. + 491. i)3-s + (6.77e3 − 6.77e3i)5-s − 4.75e4i·7-s + 3.05e5i·9-s + (3.48e5 − 3.48e5i)11-s + (6.23e5 + 6.23e5i)13-s + 6.65e6·15-s − 4.47e5·17-s + (−1.39e7 − 1.39e7i)19-s + (2.33e7 − 2.33e7i)21-s − 4.84e7i·23-s − 4.30e7i·25-s + (−6.30e7 + 6.30e7i)27-s + (−4.44e6 − 4.44e6i)29-s + 1.41e8·31-s + ⋯
L(s)  = 1  + (1.16 + 1.16i)3-s + (0.969 − 0.969i)5-s − 1.06i·7-s + 1.72i·9-s + (0.652 − 0.652i)11-s + (0.465 + 0.465i)13-s + 2.26·15-s − 0.0765·17-s + (−1.28 − 1.28i)19-s + (1.24 − 1.24i)21-s − 1.56i·23-s − 0.881i·25-s + (−0.845 + 0.845i)27-s + (−0.0402 − 0.0402i)29-s + 0.887·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.911 + 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.911 + 0.411i$
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: \((2,\ 64,\ (\ :11/2),\ 0.911 + 0.411i)\)

Particular Values

\(L(6)\) \(\approx\) \(3.81046 - 0.819658i\)
\(L(\frac12)\) \(\approx\) \(3.81046 - 0.819658i\)
\(L(\frac{13}{2})\) 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 \( 1 + (-491. - 491. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-6.77e3 + 6.77e3i)T - 4.88e7iT^{2} \)
7 \( 1 + 4.75e4iT - 1.97e9T^{2} \)
11 \( 1 + (-3.48e5 + 3.48e5i)T - 2.85e11iT^{2} \)
13 \( 1 + (-6.23e5 - 6.23e5i)T + 1.79e12iT^{2} \)
17 \( 1 + 4.47e5T + 3.42e13T^{2} \)
19 \( 1 + (1.39e7 + 1.39e7i)T + 1.16e14iT^{2} \)
23 \( 1 + 4.84e7iT - 9.52e14T^{2} \)
29 \( 1 + (4.44e6 + 4.44e6i)T + 1.22e16iT^{2} \)
31 \( 1 - 1.41e8T + 2.54e16T^{2} \)
37 \( 1 + (-4.51e7 + 4.51e7i)T - 1.77e17iT^{2} \)
41 \( 1 + 3.91e8iT - 5.50e17T^{2} \)
43 \( 1 + (8.76e8 - 8.76e8i)T - 9.29e17iT^{2} \)
47 \( 1 - 1.57e9T + 2.47e18T^{2} \)
53 \( 1 + (3.45e9 - 3.45e9i)T - 9.26e18iT^{2} \)
59 \( 1 + (-5.84e9 + 5.84e9i)T - 3.01e19iT^{2} \)
61 \( 1 + (-1.80e9 - 1.80e9i)T + 4.35e19iT^{2} \)
67 \( 1 + (-5.33e9 - 5.33e9i)T + 1.22e20iT^{2} \)
71 \( 1 - 6.24e9iT - 2.31e20T^{2} \)
73 \( 1 - 2.44e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.55e8T + 7.47e20T^{2} \)
83 \( 1 + (5.06e8 + 5.06e8i)T + 1.28e21iT^{2} \)
89 \( 1 - 4.97e10iT - 2.77e21T^{2} \)
97 \( 1 - 5.39e10T + 7.15e21T^{2} \)
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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

−12.98785193390999195371542655630, −10.98406325874568852767062746470, −10.02557144125287769883277793854, −8.994075630625398511401106919268, −8.470858958799813309526973889274, −6.49845655643452473733982606745, −4.71653901026003267007193204267, −3.99070290927026068094146805600, −2.43564881544394458264416790166, −0.899955977593718970262440866967, 1.62790296953645407816566773887, 2.24450097840942181882270226104, 3.40575706399147427769773717415, 5.90677693676648266140265476182, 6.73757164897919276919757771776, 8.021122049118582185713854977628, 9.066216483080612193225336634039, 10.15979920561716159980627195347, 11.86638017461292832542136899107, 12.88711195445633504304067929210

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