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Properties

Label 2-2e9-16.13-c1-0-9
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 $0$

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

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

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: $\chi_{512} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 512,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18874 - 0.794291i\)
\(L(\frac12)\) \(\approx\) \(1.18874 - 0.794291i\)
\(L(\frac{3}{2})\) 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

−10.52001374860430329285730054128, −9.758859936295986518668239176825, −9.196400020440862872369552590347, −7.908003555302383178461448082083, −7.26818589013450089758301648273, −5.77660705467302345136684631443, −5.36316779682565399323150483515, −3.85485476713265904517512018193, −2.69885519437037653529796435207, −0.892281956705227384142959632513, 1.85044050054771805168019821255, 2.97598496441294941622438905756, 4.51558208219417136938845621403, 5.38909910079939966506619296906, 6.54334900405889425656547510256, 7.45643673119267695566580024035, 8.233111594885068046670183703719, 9.714459441067935872421169197455, 9.896753913495995417310259114373, 11.06236775639795567100244089205

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