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Properties

Label 2-2e6-16.5-c13-0-14
Degree $2$
Conductor $64$
Sign $-0.800 - 0.599i$
Analytic cond. $68.6277$
Root an. cond. $8.28418$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−293. + 293. i)3-s + (2.27e4 + 2.27e4i)5-s + 3.43e5i·7-s + 1.42e6i·9-s + (7.55e6 + 7.55e6i)11-s + (−6.02e6 + 6.02e6i)13-s − 1.33e7·15-s + 1.22e8·17-s + (−1.66e8 + 1.66e8i)19-s + (−1.00e8 − 1.00e8i)21-s + 7.77e8i·23-s − 1.88e8i·25-s + (−8.85e8 − 8.85e8i)27-s + (3.04e9 − 3.04e9i)29-s + 6.68e9·31-s + ⋯
L(s)  = 1  + (−0.232 + 0.232i)3-s + (0.650 + 0.650i)5-s + 1.10i·7-s + 0.891i·9-s + (1.28 + 1.28i)11-s + (−0.346 + 0.346i)13-s − 0.302·15-s + 1.23·17-s + (−0.810 + 0.810i)19-s + (−0.256 − 0.256i)21-s + 1.09i·23-s − 0.154i·25-s + (−0.439 − 0.439i)27-s + (0.950 − 0.950i)29-s + 1.35·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(68.6277\)
Root analytic conductor: \(8.28418\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :13/2),\ -0.800 - 0.599i)\)

Particular Values

\(L(7)\) \(\approx\) \(2.504778056\)
\(L(\frac12)\) \(\approx\) \(2.504778056\)
\(L(\frac{15}{2})\) 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 \( 1 + (293. - 293. i)T - 1.59e6iT^{2} \)
5 \( 1 + (-2.27e4 - 2.27e4i)T + 1.22e9iT^{2} \)
7 \( 1 - 3.43e5iT - 9.68e10T^{2} \)
11 \( 1 + (-7.55e6 - 7.55e6i)T + 3.45e13iT^{2} \)
13 \( 1 + (6.02e6 - 6.02e6i)T - 3.02e14iT^{2} \)
17 \( 1 - 1.22e8T + 9.90e15T^{2} \)
19 \( 1 + (1.66e8 - 1.66e8i)T - 4.20e16iT^{2} \)
23 \( 1 - 7.77e8iT - 5.04e17T^{2} \)
29 \( 1 + (-3.04e9 + 3.04e9i)T - 1.02e19iT^{2} \)
31 \( 1 - 6.68e9T + 2.44e19T^{2} \)
37 \( 1 + (-7.06e9 - 7.06e9i)T + 2.43e20iT^{2} \)
41 \( 1 + 4.77e10iT - 9.25e20T^{2} \)
43 \( 1 + (-3.20e10 - 3.20e10i)T + 1.71e21iT^{2} \)
47 \( 1 + 4.56e10T + 5.46e21T^{2} \)
53 \( 1 + (5.36e10 + 5.36e10i)T + 2.60e22iT^{2} \)
59 \( 1 + (3.79e10 + 3.79e10i)T + 1.04e23iT^{2} \)
61 \( 1 + (-2.48e11 + 2.48e11i)T - 1.61e23iT^{2} \)
67 \( 1 + (6.55e10 - 6.55e10i)T - 5.48e23iT^{2} \)
71 \( 1 + 4.07e11iT - 1.16e24T^{2} \)
73 \( 1 - 1.83e12iT - 1.67e24T^{2} \)
79 \( 1 + 8.45e11T + 4.66e24T^{2} \)
83 \( 1 + (-2.16e11 + 2.16e11i)T - 8.87e24iT^{2} \)
89 \( 1 - 2.96e12iT - 2.19e25T^{2} \)
97 \( 1 + 6.69e12T + 6.73e25T^{2} \)
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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

−12.39004839422932992836743631455, −11.67958296452867860674168634449, −10.16329444124826851789490365976, −9.577877914211021581827346177559, −8.044956466608961801597071455185, −6.61658350576503051696724884560, −5.61314648978263902900894334419, −4.30221021956960213477365637531, −2.54208514515694648143858544199, −1.65060996471673539974896278678, 0.75917402563740850964404775313, 1.07373992199320484648290901846, 3.17707274342065498402726701892, 4.45798072117767951188746383880, 5.96910597193487729738591022436, 6.82839430654510376965528333723, 8.431401039122010655457921879259, 9.454917996345585584584926959076, 10.62078182510661573753394184661, 11.87119109114944816365503831544

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