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Properties

Label 2-2e6-16.13-c21-0-13
Degree $2$
Conductor $64$
Sign $0.637 - 0.770i$
Analytic cond. $178.865$
Root an. cond. $13.3740$
Motivic weight $21$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.29e4 + 2.29e4i)3-s + (−9.16e6 + 9.16e6i)5-s + 5.91e8i·7-s − 9.40e9i·9-s + (−4.03e10 + 4.03e10i)11-s + (−4.34e11 − 4.34e11i)13-s − 4.21e11·15-s − 3.88e12·17-s + (−9.73e12 − 9.73e12i)19-s + (−1.35e13 + 1.35e13i)21-s − 8.87e13i·23-s + 3.08e14i·25-s + (4.56e14 − 4.56e14i)27-s + (−6.29e12 − 6.29e12i)29-s + 2.43e15·31-s + ⋯
L(s)  = 1  + (0.224 + 0.224i)3-s + (−0.419 + 0.419i)5-s + 0.790i·7-s − 0.899i·9-s + (−0.468 + 0.468i)11-s + (−0.874 − 0.874i)13-s − 0.188·15-s − 0.467·17-s + (−0.364 − 0.364i)19-s + (−0.177 + 0.177i)21-s − 0.446i·23-s + 0.647i·25-s + (0.426 − 0.426i)27-s + (−0.00278 − 0.00278i)29-s + 0.534·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(178.865\)
Root analytic conductor: \(13.3740\)
Motivic weight: \(21\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :21/2),\ 0.637 - 0.770i)\)

Particular Values

\(L(11)\) \(\approx\) \(1.425832678\)
\(L(\frac12)\) \(\approx\) \(1.425832678\)
\(L(\frac{23}{2})\) 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 \( 1 + (-2.29e4 - 2.29e4i)T + 1.04e10iT^{2} \)
5 \( 1 + (9.16e6 - 9.16e6i)T - 4.76e14iT^{2} \)
7 \( 1 - 5.91e8iT - 5.58e17T^{2} \)
11 \( 1 + (4.03e10 - 4.03e10i)T - 7.40e21iT^{2} \)
13 \( 1 + (4.34e11 + 4.34e11i)T + 2.47e23iT^{2} \)
17 \( 1 + 3.88e12T + 6.90e25T^{2} \)
19 \( 1 + (9.73e12 + 9.73e12i)T + 7.14e26iT^{2} \)
23 \( 1 + 8.87e13iT - 3.94e28T^{2} \)
29 \( 1 + (6.29e12 + 6.29e12i)T + 5.13e30iT^{2} \)
31 \( 1 - 2.43e15T + 2.08e31T^{2} \)
37 \( 1 + (-2.75e16 + 2.75e16i)T - 8.55e32iT^{2} \)
41 \( 1 + 1.75e16iT - 7.38e33T^{2} \)
43 \( 1 + (-9.51e16 + 9.51e16i)T - 2.00e34iT^{2} \)
47 \( 1 + 4.70e17T + 1.30e35T^{2} \)
53 \( 1 + (1.55e18 - 1.55e18i)T - 1.62e36iT^{2} \)
59 \( 1 + (-3.87e18 + 3.87e18i)T - 1.54e37iT^{2} \)
61 \( 1 + (-2.76e18 - 2.76e18i)T + 3.10e37iT^{2} \)
67 \( 1 + (1.01e19 + 1.01e19i)T + 2.22e38iT^{2} \)
71 \( 1 - 3.67e19iT - 7.52e38T^{2} \)
73 \( 1 - 1.88e19iT - 1.34e39T^{2} \)
79 \( 1 - 2.60e19T + 7.08e39T^{2} \)
83 \( 1 + (-1.29e20 - 1.29e20i)T + 1.99e40iT^{2} \)
89 \( 1 + 4.10e20iT - 8.65e40T^{2} \)
97 \( 1 + 9.27e20T + 5.27e41T^{2} \)
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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

−11.06405169329836717125333523867, −9.911136886361158500846768620639, −8.967245383474595341581480776161, −7.81570059857422354082588359727, −6.72826807408084827870385091380, −5.49642474200175249248134381953, −4.30749404744659254743713622825, −3.05210967141066071328500047958, −2.29919621618791202717169295881, −0.58135842165466315860191784416, 0.41933996197348712697410258369, 1.65970208799785597142569587304, 2.75878512947994296644788837970, 4.20195246030375952119618859958, 4.94939763565364724525667120171, 6.51887436250870711974268602048, 7.66168484442539600255253057081, 8.339467450509307027609857525913, 9.722048264639829471014264839395, 10.79716718557945031463871905446

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