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Properties

Label 2-2e9-16.5-c1-0-10
Degree $2$
Conductor $512$
Sign $0.707 + 0.707i$
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  + (0.765 − 0.765i)3-s + (0.414 + 0.414i)5-s − 3.69i·7-s + 1.82i·9-s + (2.93 + 2.93i)11-s + (2.41 − 2.41i)13-s + 0.634·15-s − 2.82·17-s + (4.46 − 4.46i)19-s + (−2.82 − 2.82i)21-s − 6.75i·23-s − 4.65i·25-s + (3.69 + 3.69i)27-s + (−5.24 + 5.24i)29-s − 3.06·31-s + ⋯
L(s)  = 1  + (0.441 − 0.441i)3-s + (0.185 + 0.185i)5-s − 1.39i·7-s + 0.609i·9-s + (0.883 + 0.883i)11-s + (0.669 − 0.669i)13-s + 0.163·15-s − 0.685·17-s + (1.02 − 1.02i)19-s + (−0.617 − 0.617i)21-s − 1.40i·23-s − 0.931i·25-s + (0.711 + 0.711i)27-s + (−0.973 + 0.973i)29-s − 0.549·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64946 - 0.683231i\)
\(L(\frac12)\) \(\approx\) \(1.64946 - 0.683231i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.765 + 0.765i)T - 3iT^{2} \)
5 \( 1 + (-0.414 - 0.414i)T + 5iT^{2} \)
7 \( 1 + 3.69iT - 7T^{2} \)
11 \( 1 + (-2.93 - 2.93i)T + 11iT^{2} \)
13 \( 1 + (-2.41 + 2.41i)T - 13iT^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \)
23 \( 1 + 6.75iT - 23T^{2} \)
29 \( 1 + (5.24 - 5.24i)T - 29iT^{2} \)
31 \( 1 + 3.06T + 31T^{2} \)
37 \( 1 + (-6.41 - 6.41i)T + 37iT^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 + (0.765 + 0.765i)T + 43iT^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + (3.24 + 3.24i)T + 53iT^{2} \)
59 \( 1 + (0.765 + 0.765i)T + 59iT^{2} \)
61 \( 1 + (-0.757 + 0.757i)T - 61iT^{2} \)
67 \( 1 + (-1.39 + 1.39i)T - 67iT^{2} \)
71 \( 1 - 8.02iT - 71T^{2} \)
73 \( 1 - 6.48iT - 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + (9.68 - 9.68i)T - 83iT^{2} \)
89 \( 1 - 4.82iT - 89T^{2} \)
97 \( 1 - 5.17T + 97T^{2} \)
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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

−10.73978831214942496194854373986, −10.00541798476827178568967476688, −8.951905023469671061771920791010, −7.967292598862909476031998939316, −7.12046820466315484046662350156, −6.55071320892947660836160137662, −4.94008160971757658023820264924, −4.00889399982428431738996244101, −2.65771095148498799291654201971, −1.20853224460657712950663456614, 1.68858570190644246044479542165, 3.24651318719033825981817599854, 4.03168973381028160062146801999, 5.69870750685701176197041160244, 6.03312197576753986730135874467, 7.49256629637383665985240184143, 8.805956440046223198749590137101, 9.099676074516127933124001251095, 9.719996216400948676134345428179, 11.36665659105848933675346758722

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