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Properties

Label 2-2e9-16.13-c1-0-4
Degree $2$
Conductor $512$
Sign $-0.382 - 0.923i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2 + 2i)3-s + (−1 + i)5-s + 4i·7-s + 5i·9-s + (2 − 2i)11-s + (−3 − 3i)13-s − 4·15-s + (−2 − 2i)19-s + (−8 + 8i)21-s − 4i·23-s + 3i·25-s + (−4 + 4i)27-s + (3 + 3i)29-s + 8·31-s + 8·33-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)3-s + (−0.447 + 0.447i)5-s + 1.51i·7-s + 1.66i·9-s + (0.603 − 0.603i)11-s + (−0.832 − 0.832i)13-s − 1.03·15-s + (−0.458 − 0.458i)19-s + (−1.74 + 1.74i)21-s − 0.834i·23-s + 0.600i·25-s + (−0.769 + 0.769i)27-s + (0.557 + 0.557i)29-s + 1.43·31-s + 1.39·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $-0.382 - 0.923i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.382 - 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03200 + 1.54449i\)
\(L(\frac12)\) \(\approx\) \(1.03200 + 1.54449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 2i)T + 3iT^{2} \)
5 \( 1 + (1 - i)T - 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (-2 + 2i)T - 11iT^{2} \)
13 \( 1 + (3 + 3i)T + 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (2 + 2i)T + 19iT^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-3 - 3i)T + 29iT^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + (2 - 2i)T - 43iT^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-6 + 6i)T - 59iT^{2} \)
61 \( 1 + (3 + 3i)T + 61iT^{2} \)
67 \( 1 + (2 + 2i)T + 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-10 - 10i)T + 83iT^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−11.03506102078398959415191232120, −10.10662443683471923793578704840, −9.317208974386718494987251222433, −8.579281457309852325511144098014, −8.032219862017146177475715167061, −6.59417153694718370966246885287, −5.34642786189251725336081061049, −4.36116716080932502653133703863, −3.08975650707919337857830423684, −2.60152567375513468867481988589, 1.03964368906994799498656426612, 2.28322453373648801816088451916, 3.80789483671956572852543918251, 4.50465825769089052782427535100, 6.49186940732716210834184777345, 7.21195266822028914438476455736, 7.76733652253301107098772646183, 8.643084682641881530707278093552, 9.585770316199146499807467568905, 10.46582720824680621975821340463

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