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Properties

Label 2-2e6-8.5-c11-0-18
Degree $2$
Conductor $64$
Sign $-0.258 + 0.965i$
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  + 182. i·3-s − 1.16e4i·5-s + 2.65e4·7-s + 1.43e5·9-s − 1.03e6i·11-s + 1.67e6i·13-s + 2.13e6·15-s + 6.69e6·17-s − 4.65e6i·19-s + 4.83e6i·21-s − 4.44e7·23-s − 8.78e7·25-s + 5.85e7i·27-s − 5.59e7i·29-s + 3.48e7·31-s + ⋯
L(s)  = 1  + 0.433i·3-s − 1.67i·5-s + 0.596·7-s + 0.812·9-s − 1.93i·11-s + 1.24i·13-s + 0.725·15-s + 1.14·17-s − 0.431i·19-s + 0.258i·21-s − 1.44·23-s − 1.79·25-s + 0.785i·27-s − 0.506i·29-s + 0.218·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.29361 - 1.68587i\)
\(L(\frac12)\) \(\approx\) \(1.29361 - 1.68587i\)
\(L(\frac{13}{2})\) 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 \( 1 - 182. iT - 1.77e5T^{2} \)
5 \( 1 + 1.16e4iT - 4.88e7T^{2} \)
7 \( 1 - 2.65e4T + 1.97e9T^{2} \)
11 \( 1 + 1.03e6iT - 2.85e11T^{2} \)
13 \( 1 - 1.67e6iT - 1.79e12T^{2} \)
17 \( 1 - 6.69e6T + 3.42e13T^{2} \)
19 \( 1 + 4.65e6iT - 1.16e14T^{2} \)
23 \( 1 + 4.44e7T + 9.52e14T^{2} \)
29 \( 1 + 5.59e7iT - 1.22e16T^{2} \)
31 \( 1 - 3.48e7T + 2.54e16T^{2} \)
37 \( 1 - 1.45e8iT - 1.77e17T^{2} \)
41 \( 1 - 9.44e8T + 5.50e17T^{2} \)
43 \( 1 + 3.43e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.06e9T + 2.47e18T^{2} \)
53 \( 1 + 3.78e9iT - 9.26e18T^{2} \)
59 \( 1 + 4.97e9iT - 3.01e19T^{2} \)
61 \( 1 + 2.99e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.63e10iT - 1.22e20T^{2} \)
71 \( 1 + 2.22e10T + 2.31e20T^{2} \)
73 \( 1 + 5.55e9T + 3.13e20T^{2} \)
79 \( 1 + 4.18e10T + 7.47e20T^{2} \)
83 \( 1 - 2.07e10iT - 1.28e21T^{2} \)
89 \( 1 - 1.82e10T + 2.77e21T^{2} \)
97 \( 1 - 1.50e8T + 7.15e21T^{2} \)
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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

−12.21268389934623063039359106728, −11.29333773681159538494730463820, −9.782333693585398489652029170274, −8.800056323147645274213068146638, −7.925361176746629231260374114200, −5.94281876884707996315216449661, −4.79875966208808573175019171303, −3.82998338301769587458807340854, −1.60479276224635332596128002374, −0.59104570729736985554557206398, 1.51794420204169876052835765735, 2.66650472482080219779115177180, 4.18867281375487821108372694738, 5.92273308458894901368237758160, 7.36680004911639922556833344054, 7.63432028230115697835909576860, 10.03175409298832511535852238667, 10.36728991996700880369957527186, 11.91360959186307310053812075943, 12.78472689640140301580562267016

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