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Properties

Label 2-2e6-16.5-c11-0-17
Degree $2$
Conductor $64$
Sign $-0.563 + 0.825i$
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  + (−13.6 + 13.6i)3-s + (−3.50e3 − 3.50e3i)5-s − 3.76e4i·7-s + 1.76e5i·9-s + (2.83e5 + 2.83e5i)11-s + (1.70e6 − 1.70e6i)13-s + 9.52e4·15-s + 1.57e6·17-s + (−1.25e7 + 1.25e7i)19-s + (5.12e5 + 5.12e5i)21-s − 6.83e6i·23-s − 2.43e7i·25-s + (−4.81e6 − 4.81e6i)27-s + (5.00e7 − 5.00e7i)29-s − 7.19e7·31-s + ⋯
L(s)  = 1  + (−0.0323 + 0.0323i)3-s + (−0.501 − 0.501i)5-s − 0.846i·7-s + 0.997i·9-s + (0.530 + 0.530i)11-s + (1.27 − 1.27i)13-s + 0.0323·15-s + 0.269·17-s + (−1.15 + 1.15i)19-s + (0.0273 + 0.0273i)21-s − 0.221i·23-s − 0.497i·25-s + (−0.0645 − 0.0645i)27-s + (0.452 − 0.452i)29-s − 0.451·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-0.563 + 0.825i$
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: \((2,\ 64,\ (\ :11/2),\ -0.563 + 0.825i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.559845 - 1.06004i\)
\(L(\frac12)\) \(\approx\) \(0.559845 - 1.06004i\)
\(L(\frac{13}{2})\) 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 \( 1 + (13.6 - 13.6i)T - 1.77e5iT^{2} \)
5 \( 1 + (3.50e3 + 3.50e3i)T + 4.88e7iT^{2} \)
7 \( 1 + 3.76e4iT - 1.97e9T^{2} \)
11 \( 1 + (-2.83e5 - 2.83e5i)T + 2.85e11iT^{2} \)
13 \( 1 + (-1.70e6 + 1.70e6i)T - 1.79e12iT^{2} \)
17 \( 1 - 1.57e6T + 3.42e13T^{2} \)
19 \( 1 + (1.25e7 - 1.25e7i)T - 1.16e14iT^{2} \)
23 \( 1 + 6.83e6iT - 9.52e14T^{2} \)
29 \( 1 + (-5.00e7 + 5.00e7i)T - 1.22e16iT^{2} \)
31 \( 1 + 7.19e7T + 2.54e16T^{2} \)
37 \( 1 + (-9.81e7 - 9.81e7i)T + 1.77e17iT^{2} \)
41 \( 1 + 7.24e8iT - 5.50e17T^{2} \)
43 \( 1 + (1.93e8 + 1.93e8i)T + 9.29e17iT^{2} \)
47 \( 1 + 1.39e9T + 2.47e18T^{2} \)
53 \( 1 + (3.32e9 + 3.32e9i)T + 9.26e18iT^{2} \)
59 \( 1 + (5.46e9 + 5.46e9i)T + 3.01e19iT^{2} \)
61 \( 1 + (4.88e9 - 4.88e9i)T - 4.35e19iT^{2} \)
67 \( 1 + (-7.47e9 + 7.47e9i)T - 1.22e20iT^{2} \)
71 \( 1 + 2.58e10iT - 2.31e20T^{2} \)
73 \( 1 + 1.88e10iT - 3.13e20T^{2} \)
79 \( 1 + 3.09e10T + 7.47e20T^{2} \)
83 \( 1 + (2.96e10 - 2.96e10i)T - 1.28e21iT^{2} \)
89 \( 1 + 6.42e10iT - 2.77e21T^{2} \)
97 \( 1 - 8.87e10T + 7.15e21T^{2} \)
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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

−12.34274275447855321188595949110, −10.90766410663934174286430181745, −10.18919256260816445007439131925, −8.439401070388297273996429153589, −7.74368955946598429802099869420, −6.16636982603523665013838142407, −4.66147445001041246023990201382, −3.61235221407378427542789626731, −1.66317643226310968980385569175, −0.34166856284617443409050887761, 1.33459843926756165904464239069, 3.04689597619773982438815309699, 4.17874853657767559236308528644, 6.05760049047698049649430756762, 6.84498599675768253387897502144, 8.609023180291709437588266832555, 9.271813807149063761835112100104, 11.10904785279969990418634958674, 11.64461849367177398203413146612, 12.88223644578421985738074349364

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