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Properties

Label 2-2e9-16.5-c1-0-14
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3 − 3i)5-s + 3i·9-s + (−1 + i)13-s − 8·17-s + 13i·25-s + (−7 + 7i)29-s + (−5 − 5i)37-s − 8i·41-s + (9 − 9i)45-s + 7·49-s + (−5 − 5i)53-s + (−1 + i)61-s + 6·65-s − 6i·73-s − 9·81-s + ⋯
L(s)  = 1  + (−1.34 − 1.34i)5-s + i·9-s + (−0.277 + 0.277i)13-s − 1.94·17-s + 2.60i·25-s + (−1.29 + 1.29i)29-s + (−0.821 − 0.821i)37-s − 1.24i·41-s + (1.34 − 1.34i)45-s + 49-s + (−0.686 − 0.686i)53-s + (−0.128 + 0.128i)61-s + 0.744·65-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: \(1\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ -0.923 - 0.382i)\)

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 \( 1 - 3iT^{2} \)
5 \( 1 + (3 + 3i)T + 5iT^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11iT^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + 8T + 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (7 - 7i)T - 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + (1 - i)T - 61iT^{2} \)
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

−10.68086206437723389100626391399, −9.069540656469781468452912457701, −8.763736629208434220392247529823, −7.71844317111024991511995539038, −7.03123875703169337427194047700, −5.35429359836309575592434763750, −4.64151812143918439234263580506, −3.78391857123772335592102552475, −1.94235188945846926758769502317, 0, 2.57789197994211068748399544012, 3.63749608110412046786385718474, 4.44215588362433810879658579021, 6.20005292841791693694369192528, 6.86681513196180977535606462490, 7.66700851176729134844715704013, 8.632234507736590101961628433772, 9.694846797595585628577505532914, 10.72756617107388021907404876625

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