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Properties

Label 2-2e6-8.5-c11-0-9
Degree $2$
Conductor $64$
Sign $0.965 - 0.258i$
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  + 803. i·3-s − 9.05e3i·5-s − 7.18e4·7-s − 4.69e5·9-s − 4.76e4i·11-s + 7.87e5i·13-s + 7.27e6·15-s − 5.65e6·17-s + 1.80e6i·19-s − 5.77e7i·21-s + 3.98e7·23-s − 3.30e7·25-s − 2.34e8i·27-s − 1.23e7i·29-s + 1.87e8·31-s + ⋯
L(s)  = 1  + 1.91i·3-s − 1.29i·5-s − 1.61·7-s − 2.64·9-s − 0.0892i·11-s + 0.588i·13-s + 2.47·15-s − 0.965·17-s + 0.167i·19-s − 3.08i·21-s + 1.29·23-s − 0.677·25-s − 3.14i·27-s − 0.112i·29-s + 1.17·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.965 - 0.258i$
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: \((2,\ 64,\ (\ :11/2),\ 0.965 - 0.258i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.00650 + 0.132508i\)
\(L(\frac12)\) \(\approx\) \(1.00650 + 0.132508i\)
\(L(\frac{13}{2})\) 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 \( 1 - 803. iT - 1.77e5T^{2} \)
5 \( 1 + 9.05e3iT - 4.88e7T^{2} \)
7 \( 1 + 7.18e4T + 1.97e9T^{2} \)
11 \( 1 + 4.76e4iT - 2.85e11T^{2} \)
13 \( 1 - 7.87e5iT - 1.79e12T^{2} \)
17 \( 1 + 5.65e6T + 3.42e13T^{2} \)
19 \( 1 - 1.80e6iT - 1.16e14T^{2} \)
23 \( 1 - 3.98e7T + 9.52e14T^{2} \)
29 \( 1 + 1.23e7iT - 1.22e16T^{2} \)
31 \( 1 - 1.87e8T + 2.54e16T^{2} \)
37 \( 1 - 1.60e8iT - 1.77e17T^{2} \)
41 \( 1 - 3.64e8T + 5.50e17T^{2} \)
43 \( 1 - 2.64e8iT - 9.29e17T^{2} \)
47 \( 1 - 4.74e8T + 2.47e18T^{2} \)
53 \( 1 + 1.34e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.70e9iT - 3.01e19T^{2} \)
61 \( 1 + 4.97e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.61e10iT - 1.22e20T^{2} \)
71 \( 1 - 1.83e10T + 2.31e20T^{2} \)
73 \( 1 + 3.02e10T + 3.13e20T^{2} \)
79 \( 1 + 3.11e10T + 7.47e20T^{2} \)
83 \( 1 - 4.96e10iT - 1.28e21T^{2} \)
89 \( 1 - 2.03e10T + 2.77e21T^{2} \)
97 \( 1 - 8.44e10T + 7.15e21T^{2} \)
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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

−12.64809853989193112262844737435, −11.32569677369469339959388146647, −10.06757264930839402310674566836, −9.271062597882000785700859899874, −8.697011528064326129223997697617, −6.30923893493163276871245836204, −5.00290400652542698187310047072, −4.12960100819216882326543492363, −2.94653318299634403055247921668, −0.40903980408126673589052354103, 0.72482704355819193417892906294, 2.49486998110374112235047301904, 3.09951320969690271867446877210, 6.00211623397750546473005507545, 6.74948603927352357367817965157, 7.35068399375724244443594527371, 8.887605402756594419547466096854, 10.48311012554582457057585757095, 11.63269177113786776526325125361, 12.85749164719511640075769914740

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