Properties

Label 2-2e9-8.5-c1-0-12
Degree $2$
Conductor $512$
Sign $i$
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  + 1.41i·3-s − 2.82i·5-s − 4·7-s + 0.999·9-s − 1.41i·11-s − 2.82i·13-s + 4.00·15-s − 4·17-s − 7.07i·19-s − 5.65i·21-s − 4·23-s − 3.00·25-s + 5.65i·27-s − 8.48i·29-s + 8·31-s + ⋯
L(s)  = 1  + 0.816i·3-s − 1.26i·5-s − 1.51·7-s + 0.333·9-s − 0.426i·11-s − 0.784i·13-s + 1.03·15-s − 0.970·17-s − 1.62i·19-s − 1.23i·21-s − 0.834·23-s − 0.600·25-s + 1.08i·27-s − 1.57i·29-s + 1.43·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $i$
Analytic conductor: \(4.08834\)
Root analytic conductor: \(2.02196\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{512} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637339 - 0.637339i\)
\(L(\frac12)\) \(\approx\) \(0.637339 - 0.637339i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 3T^{2} \)
5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 + 4T + 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 4T + 17T^{2} \)
19 \( 1 + 7.07iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 8.48iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2.82iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + 4.24iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−10.39017818164891154948924488157, −9.739293793711422052090321075488, −9.074550109121390332568244153649, −8.289653010754335617108701445613, −6.89615184515577686861177937745, −5.92858865262993170321113365516, −4.80370649194988237267841040793, −4.04997260810033621865495591668, −2.78191032351616055652616608942, −0.51977093546814629137452395290, 1.92373065670686808966821961592, 3.10550810409748815510461585071, 4.17305747562492982760510545305, 6.06050828383155309270346852197, 6.71202377626927932295426514956, 7.09621972605103215146707564603, 8.275633765158505299243361421439, 9.644120037497726486199151912738, 10.10902537489330026071095475360, 11.06667657864456293587325431395

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