Properties

Label 2-2e6-16.13-c3-0-3
Degree $2$
Conductor $64$
Sign $0.659 + 0.751i$
Analytic cond. $3.77612$
Root an. cond. $1.94322$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.756 − 0.756i)3-s + (8.22 − 8.22i)5-s + 2.67i·7-s − 25.8i·9-s + (45.2 − 45.2i)11-s + (35.3 + 35.3i)13-s − 12.4·15-s − 72.4·17-s + (−19.4 − 19.4i)19-s + (2.02 − 2.02i)21-s + 139. i·23-s − 10.3i·25-s + (−39.9 + 39.9i)27-s + (66.0 + 66.0i)29-s − 188.·31-s + ⋯
L(s)  = 1  + (−0.145 − 0.145i)3-s + (0.735 − 0.735i)5-s + 0.144i·7-s − 0.957i·9-s + (1.23 − 1.23i)11-s + (0.755 + 0.755i)13-s − 0.214·15-s − 1.03·17-s + (−0.234 − 0.234i)19-s + (0.0210 − 0.0210i)21-s + 1.26i·23-s − 0.0826i·25-s + (−0.285 + 0.285i)27-s + (0.422 + 0.422i)29-s − 1.09·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.659 + 0.751i$
Analytic conductor: \(3.77612\)
Root analytic conductor: \(1.94322\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :3/2),\ 0.659 + 0.751i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.40575 - 0.636791i\)
\(L(\frac12)\) \(\approx\) \(1.40575 - 0.636791i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.756 + 0.756i)T + 27iT^{2} \)
5 \( 1 + (-8.22 + 8.22i)T - 125iT^{2} \)
7 \( 1 - 2.67iT - 343T^{2} \)
11 \( 1 + (-45.2 + 45.2i)T - 1.33e3iT^{2} \)
13 \( 1 + (-35.3 - 35.3i)T + 2.19e3iT^{2} \)
17 \( 1 + 72.4T + 4.91e3T^{2} \)
19 \( 1 + (19.4 + 19.4i)T + 6.85e3iT^{2} \)
23 \( 1 - 139. iT - 1.21e4T^{2} \)
29 \( 1 + (-66.0 - 66.0i)T + 2.43e4iT^{2} \)
31 \( 1 + 188.T + 2.97e4T^{2} \)
37 \( 1 + (84.0 - 84.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 104. iT - 6.89e4T^{2} \)
43 \( 1 + (-31.4 + 31.4i)T - 7.95e4iT^{2} \)
47 \( 1 - 488.T + 1.03e5T^{2} \)
53 \( 1 + (-149. + 149. i)T - 1.48e5iT^{2} \)
59 \( 1 + (284. - 284. i)T - 2.05e5iT^{2} \)
61 \( 1 + (228. + 228. i)T + 2.26e5iT^{2} \)
67 \( 1 + (139. + 139. i)T + 3.00e5iT^{2} \)
71 \( 1 - 453. iT - 3.57e5T^{2} \)
73 \( 1 + 259. iT - 3.89e5T^{2} \)
79 \( 1 + 323.T + 4.93e5T^{2} \)
83 \( 1 + (-563. - 563. i)T + 5.71e5iT^{2} \)
89 \( 1 - 866. iT - 7.04e5T^{2} \)
97 \( 1 + 936.T + 9.12e5T^{2} \)
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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

−14.03803881646030885499966859249, −13.29643086479526408918439998813, −12.00570296419898508471957167495, −11.08552667066933640568122690095, −9.139047392768155405288097559397, −8.945298717187409178857270270335, −6.69360372294776558653826170350, −5.73057989129939700826834027815, −3.83257262799237546555828531932, −1.29966905722554692889204139218, 2.16938548786308017743727169084, 4.32866550440048001640238788922, 6.05060583281485487136999674180, 7.19468984980944091242357768239, 8.850782256709264084158188553961, 10.23440572983994847269183129409, 10.89381050206703575411626484095, 12.40907741767964524979904886773, 13.61479514163115188385251951888, 14.49758831149885971916281093668

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