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Properties

Label 2-2e9-16.13-c1-0-7
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  + (1.84 + 1.84i)3-s + (2.41 − 2.41i)5-s + 1.53i·7-s + 3.82i·9-s + (−3.37 + 3.37i)11-s + (0.414 + 0.414i)13-s + 8.92·15-s + 2.82·17-s + (0.317 + 0.317i)19-s + (−2.82 + 2.82i)21-s − 5.86i·23-s − 6.65i·25-s + (−1.53 + 1.53i)27-s + (−3.24 − 3.24i)29-s + 7.39·31-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)3-s + (1.07 − 1.07i)5-s + 0.578i·7-s + 1.27i·9-s + (−1.01 + 1.01i)11-s + (0.114 + 0.114i)13-s + 2.30·15-s + 0.685·17-s + (0.0727 + 0.0727i)19-s + (−0.617 + 0.617i)21-s − 1.22i·23-s − 1.33i·25-s + (−0.294 + 0.294i)27-s + (−0.602 − 0.602i)29-s + 1.32·31-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: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11205 + 0.874839i\)
\(L(\frac12)\) \(\approx\) \(2.11205 + 0.874839i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 - 1.84i)T + 3iT^{2} \)
5 \( 1 + (-2.41 + 2.41i)T - 5iT^{2} \)
7 \( 1 - 1.53iT - 7T^{2} \)
11 \( 1 + (3.37 - 3.37i)T - 11iT^{2} \)
13 \( 1 + (-0.414 - 0.414i)T + 13iT^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 + (-0.317 - 0.317i)T + 19iT^{2} \)
23 \( 1 + 5.86iT - 23T^{2} \)
29 \( 1 + (3.24 + 3.24i)T + 29iT^{2} \)
31 \( 1 - 7.39T + 31T^{2} \)
37 \( 1 + (3.58 - 3.58i)T - 37iT^{2} \)
41 \( 1 + 4iT - 41T^{2} \)
43 \( 1 + (1.84 - 1.84i)T - 43iT^{2} \)
47 \( 1 + 7.39T + 47T^{2} \)
53 \( 1 + (5.24 - 5.24i)T - 53iT^{2} \)
59 \( 1 + (1.84 - 1.84i)T - 59iT^{2} \)
61 \( 1 + (9.24 + 9.24i)T + 61iT^{2} \)
67 \( 1 + (7.07 + 7.07i)T + 67iT^{2} \)
71 \( 1 + 11.9iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 6.12T + 79T^{2} \)
83 \( 1 + (2.48 + 2.48i)T + 83iT^{2} \)
89 \( 1 - 0.828iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.48050516469890782476421576160, −9.928334294965333555742261260004, −9.309201589082984099557934053831, −8.577005000644167465094694793582, −7.82254255919047276277555132864, −6.15359392522684595643942538516, −5.04502765193822303228489332363, −4.52100256636222113202184407758, −2.95660436528853061084817827584, −1.95666886106626110071971252243, 1.50307546148275386429436592195, 2.76122460026501772446708132863, 3.35455143084992229620120561989, 5.45145363565279620900635943970, 6.35946214521999873930923891196, 7.29940106536115718693571368431, 7.891584474528461407573362145097, 8.891812622466073261497008566234, 9.977978395772001444740741766578, 10.60133042273829085990092265662

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