Properties

Label 2-2e6-16.5-c11-0-8
Degree $2$
Conductor $64$
Sign $0.130 - 0.991i$
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  + (−137. + 137. i)3-s + (3.89e3 + 3.89e3i)5-s − 3.79e4i·7-s + 1.39e5i·9-s + (6.81e4 + 6.81e4i)11-s + (−1.09e5 + 1.09e5i)13-s − 1.06e6·15-s + 8.85e6·17-s + (7.12e5 − 7.12e5i)19-s + (5.19e6 + 5.19e6i)21-s − 2.69e7i·23-s − 1.84e7i·25-s + (−4.34e7 − 4.34e7i)27-s + (−1.10e8 + 1.10e8i)29-s + 1.15e8·31-s + ⋯
L(s)  = 1  + (−0.325 + 0.325i)3-s + (0.558 + 0.558i)5-s − 0.853i·7-s + 0.787i·9-s + (0.127 + 0.127i)11-s + (−0.0820 + 0.0820i)13-s − 0.363·15-s + 1.51·17-s + (0.0660 − 0.0660i)19-s + (0.277 + 0.277i)21-s − 0.873i·23-s − 0.377i·25-s + (−0.582 − 0.582i)27-s + (−0.999 + 0.999i)29-s + 0.725·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $0.130 - 0.991i$
Analytic conductor: \(49.1739\)
Root analytic conductor: \(7.01241\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{64} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 64,\ (\ :11/2),\ 0.130 - 0.991i)\)

Particular Values

\(L(6)\) \(\approx\) \(1.44638 + 1.26792i\)
\(L(\frac12)\) \(\approx\) \(1.44638 + 1.26792i\)
\(L(\frac{13}{2})\) 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 \( 1 + (137. - 137. i)T - 1.77e5iT^{2} \)
5 \( 1 + (-3.89e3 - 3.89e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 3.79e4iT - 1.97e9T^{2} \)
11 \( 1 + (-6.81e4 - 6.81e4i)T + 2.85e11iT^{2} \)
13 \( 1 + (1.09e5 - 1.09e5i)T - 1.79e12iT^{2} \)
17 \( 1 - 8.85e6T + 3.42e13T^{2} \)
19 \( 1 + (-7.12e5 + 7.12e5i)T - 1.16e14iT^{2} \)
23 \( 1 + 2.69e7iT - 9.52e14T^{2} \)
29 \( 1 + (1.10e8 - 1.10e8i)T - 1.22e16iT^{2} \)
31 \( 1 - 1.15e8T + 2.54e16T^{2} \)
37 \( 1 + (-3.15e8 - 3.15e8i)T + 1.77e17iT^{2} \)
41 \( 1 - 8.66e8iT - 5.50e17T^{2} \)
43 \( 1 + (-3.13e8 - 3.13e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.93e9T + 2.47e18T^{2} \)
53 \( 1 + (-8.24e8 - 8.24e8i)T + 9.26e18iT^{2} \)
59 \( 1 + (-7.47e9 - 7.47e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (6.44e8 - 6.44e8i)T - 4.35e19iT^{2} \)
67 \( 1 + (5.39e9 - 5.39e9i)T - 1.22e20iT^{2} \)
71 \( 1 - 2.90e10iT - 2.31e20T^{2} \)
73 \( 1 - 1.12e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.03e7T + 7.47e20T^{2} \)
83 \( 1 + (2.10e10 - 2.10e10i)T - 1.28e21iT^{2} \)
89 \( 1 - 7.01e10iT - 2.77e21T^{2} \)
97 \( 1 + 3.85e10T + 7.15e21T^{2} \)
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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

−12.97169791474668380435440448145, −11.52257418954465587675261326876, −10.42057123858679923786160598809, −9.874012225262094525263625818036, −8.074356451889728857419597470267, −6.90316963175666317279385887078, −5.59020168696249551816236418914, −4.30197579967325508197686241829, −2.75162440726841697158098683790, −1.16525965869275354113857897482, 0.60646832247725134795813577751, 1.82949176811709726442178600738, 3.48876628635047992667323694603, 5.37089479951330280681227041026, 6.05131394448565169253598588875, 7.62498833162292018637528442496, 9.047845745117391054202977666452, 9.816840481935231902915643649931, 11.53993376108854701179429205455, 12.30651984958401377103830920301

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