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Properties

Label 2-832-104.101-c1-0-1
Degree $2$
Conductor $832$
Sign $-0.308 - 0.951i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
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  + (0.633 − 0.366i)3-s − 1.73·5-s + (−1.73 − i)7-s + (−1.23 + 2.13i)9-s + (1.73 + 3i)11-s + (1.59 − 3.23i)13-s + (−1.09 + 0.633i)15-s + (−3.23 + 5.59i)17-s + (0.633 − 1.09i)19-s − 1.46·21-s + (0.633 + 1.09i)23-s − 2.00·25-s + 4i·27-s + (−5.59 + 3.23i)29-s + 10.1i·31-s + ⋯
L(s)  = 1  + (0.366 − 0.211i)3-s − 0.774·5-s + (−0.654 − 0.377i)7-s + (−0.410 + 0.711i)9-s + (0.522 + 0.904i)11-s + (0.443 − 0.896i)13-s + (−0.283 + 0.163i)15-s + (−0.783 + 1.35i)17-s + (0.145 − 0.251i)19-s − 0.319·21-s + (0.132 + 0.228i)23-s − 0.400·25-s + 0.769i·27-s + (−1.03 + 0.600i)29-s + 1.83i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.308 - 0.951i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.308 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.471656 + 0.648647i\)
\(L(\frac12)\) \(\approx\) \(0.471656 + 0.648647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-1.59 + 3.23i)T \)
good3 \( 1 + (-0.633 + 0.366i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.73T + 5T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.59 - 3.23i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + (0.598 + 1.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.26 - 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.66iT - 47T^{2} \)
53 \( 1 - 5.53iT - 53T^{2} \)
59 \( 1 + (-6.46 + 11.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.40 + 3.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.09 + 1.90i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.90 + 2.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.19iT - 73T^{2} \)
79 \( 1 - 6T + 79T^{2} \)
83 \( 1 - 9.46T + 83T^{2} \)
89 \( 1 + (-7.39 + 4.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 + 3i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65679494974465807912135371129, −9.557680948120433187923311349020, −8.653684458413111282955743035154, −7.913715352367541783703451502710, −7.17247970340504704018879209695, −6.27071918000224318242778360308, −5.06224650750426655402974420933, −3.94786455650009749135626388239, −3.14921534986322303045871277928, −1.68689780505711253617367394809, 0.36881444338327829491246210731, 2.47934172792982154625167041294, 3.61045688886475114760253619558, 4.16366816529842385055638701453, 5.71267156591568723983466626995, 6.45633425979588760882457860160, 7.39893191148185263326444071232, 8.476254573469116374815935146526, 9.169579113922946061716344237283, 9.589136774981603846218848272997

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