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Properties

Label 2-2e9-32.13-c1-0-9
Degree $2$
Conductor $512$
Sign $0.453 + 0.891i$
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  + (0.529 − 1.27i)3-s + (1.70 − 0.707i)5-s + (2.74 − 2.74i)7-s + (0.766 + 0.766i)9-s + (0.0560 + 0.135i)11-s + (−2.85 − 1.18i)13-s − 2.55i·15-s + 6.44i·17-s + (−1.94 − 0.805i)19-s + (−2.05 − 4.97i)21-s + (−0.749 − 0.749i)23-s + (−1.12 + 1.12i)25-s + (5.22 − 2.16i)27-s + (1.79 − 4.32i)29-s + 1.17·31-s + ⋯
L(s)  = 1  + (0.305 − 0.738i)3-s + (0.763 − 0.316i)5-s + (1.03 − 1.03i)7-s + (0.255 + 0.255i)9-s + (0.0169 + 0.0408i)11-s + (−0.790 − 0.327i)13-s − 0.660i·15-s + 1.56i·17-s + (−0.445 − 0.184i)19-s + (−0.449 − 1.08i)21-s + (−0.156 − 0.156i)23-s + (−0.224 + 0.224i)25-s + (1.00 − 0.416i)27-s + (0.332 − 0.802i)29-s + 0.210·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.453 + 0.891i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.453 + 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66112 - 1.01893i\)
\(L(\frac12)\) \(\approx\) \(1.66112 - 1.01893i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.529 + 1.27i)T + (-2.12 - 2.12i)T^{2} \)
5 \( 1 + (-1.70 + 0.707i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.74 + 2.74i)T - 7iT^{2} \)
11 \( 1 + (-0.0560 - 0.135i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.85 + 1.18i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 6.44iT - 17T^{2} \)
19 \( 1 + (1.94 + 0.805i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.749 + 0.749i)T + 23iT^{2} \)
29 \( 1 + (-1.79 + 4.32i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (4.18 - 1.73i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-2.49 - 2.49i)T + 41iT^{2} \)
43 \( 1 + (2.52 + 6.10i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 2.66iT - 47T^{2} \)
53 \( 1 + (-0.682 - 1.64i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (3.47 - 1.43i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.43 - 3.46i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (5.83 - 14.0i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-3.40 + 3.40i)T - 71iT^{2} \)
73 \( 1 + (-0.442 - 0.442i)T + 73iT^{2} \)
79 \( 1 - 7.07iT - 79T^{2} \)
83 \( 1 + (7.23 + 2.99i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.21 + 4.21i)T - 89iT^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−10.44561006845486133314672750111, −10.17340929237113067830130179532, −8.733699764940999325323058414117, −7.949964404668675430889270744393, −7.28939199035666584035811935991, −6.21639847144759890329572477788, −5.02119890377441557779432727332, −4.08712587760529242563133046164, −2.23158340691775057509727742936, −1.34397974764690524523124868783, 1.94962413249165835278545105062, 3.00234853888873776465978660604, 4.57023994977045490777977992854, 5.21566656386872709880307208529, 6.37772455460444876124587668005, 7.47879846873068311172018421195, 8.651879904424871914441495954426, 9.363749301165347314606441331434, 9.973244148684431073807179576031, 10.96860603942781559593329111530

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