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Properties

Label 2-2e9-16.5-c1-0-8
Degree $2$
Conductor $512$
Sign $-0.382 + 0.923i$
Analytic cond. $4.08834$
Root an. cond. $2.02196$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + 2i)3-s + (−1 − i)5-s + 4i·7-s − 5i·9-s + (−2 − 2i)11-s + (−3 + 3i)13-s + 4·15-s + (2 − 2i)19-s + (−8 − 8i)21-s − 4i·23-s − 3i·25-s + (4 + 4i)27-s + (3 − 3i)29-s − 8·31-s + 8·33-s + ⋯
L(s)  = 1  + (−1.15 + 1.15i)3-s + (−0.447 − 0.447i)5-s + 1.51i·7-s − 1.66i·9-s + (−0.603 − 0.603i)11-s + (−0.832 + 0.832i)13-s + 1.03·15-s + (0.458 − 0.458i)19-s + (−1.74 − 1.74i)21-s − 0.834i·23-s − 0.600i·25-s + (0.769 + 0.769i)27-s + (0.557 − 0.557i)29-s − 1.43·31-s + 1.39·33-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 2i)T - 3iT^{2} \)
5 \( 1 + (1 + i)T + 5iT^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + (2 + 2i)T + 11iT^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (-2 + 2i)T - 19iT^{2} \)
23 \( 1 + 4iT - 23T^{2} \)
29 \( 1 + (-3 + 3i)T - 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-1 - i)T + 37iT^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (6 + 6i)T + 59iT^{2} \)
61 \( 1 + (3 - 3i)T - 61iT^{2} \)
67 \( 1 + (-2 + 2i)T - 67iT^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (10 - 10i)T - 83iT^{2} \)
89 \( 1 - 14iT - 89T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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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

−10.75984207574315568072174627567, −9.708975134850384076379739323305, −9.062956555453450135335579972865, −8.162836717193120410632572548166, −6.65108369111800054916211922486, −5.61144621036201557340479689475, −5.05923233073200046327859911290, −4.14789643453972268610350955997, −2.60915342710743998365373176195, 0, 1.47178569306772076606250035448, 3.28821402933475174139413419573, 4.74470352848300464199515850385, 5.67108331264863843287405349502, 6.88982122500765121758887663224, 7.46639080553063419008966883211, 7.79552032251226165920874563536, 9.747173572013814564060513999356, 10.58531895000847694922554123133

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