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Properties

Label 2-2e9-16.5-c1-0-6
Degree $2$
Conductor $512$
Sign $0.923 + 0.382i$
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 − i)5-s + 3i·9-s + (5 − 5i)13-s + 8·17-s − 3i·25-s + (3 − 3i)29-s + (−7 − 7i)37-s + 8i·41-s + (3 − 3i)45-s + 7·49-s + (9 + 9i)53-s + (−11 + 11i)61-s − 10·65-s − 6i·73-s − 9·81-s + ⋯
L(s)  = 1  + (−0.447 − 0.447i)5-s + i·9-s + (1.38 − 1.38i)13-s + 1.94·17-s − 0.600i·25-s + (0.557 − 0.557i)29-s + (−1.15 − 1.15i)37-s + 1.24i·41-s + (0.447 − 0.447i)45-s + 49-s + (1.23 + 1.23i)53-s + (−1.40 + 1.40i)61-s − 1.24·65-s − 0.702i·73-s − 81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(512\)    =    \(2^{9}\)
Sign: $0.923 + 0.382i$
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: \((2,\ 512,\ (\ :1/2),\ 0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40221 - 0.278918i\)
\(L(\frac12)\) \(\approx\) \(1.40221 - 0.278918i\)
\(L(\frac{3}{2})\) 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 \( 1 - 3iT^{2} \)
5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + (-5 + 5i)T - 13iT^{2} \)
17 \( 1 - 8T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (7 + 7i)T + 37iT^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-9 - 9i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (11 - 11i)T - 61iT^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.59295778641146198552345709771, −10.28848396900697739634015199402, −8.866597166253576373314699495777, −8.048848208304325473870910289010, −7.57355083188020345453515884079, −5.97668910654611376423357860269, −5.30213321231904925464493723223, −4.05222104021183707949451473760, −2.92793138425852203606671078724, −1.09270423426689080323935150257, 1.34762916831382432078830694610, 3.31615130913095394623605156135, 3.89294201188254270664626340478, 5.42686253922564345940117789280, 6.48900057728029578050946308496, 7.17765136673761200507377119936, 8.361559591341289211257769191423, 9.133028327215641590068972868978, 10.08246970809745146308831262477, 11.02386975674152690467768652483

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