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Properties

Label 2-832-104.101-c1-0-17
Degree $2$
Conductor $832$
Sign $-0.759 + 0.650i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
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  + (−2.36 + 1.36i)3-s + 1.73·5-s + (−1.73 − i)7-s + (2.23 − 3.86i)9-s + (1.73 + 3i)11-s + (−3.59 − 0.232i)13-s + (−4.09 + 2.36i)15-s + (0.232 − 0.401i)17-s + (−2.36 + 4.09i)19-s + 5.46·21-s + (−2.36 − 4.09i)23-s − 2.00·25-s + 4.00i·27-s + (−0.401 + 0.232i)29-s − 0.196i·31-s + ⋯
L(s)  = 1  + (−1.36 + 0.788i)3-s + 0.774·5-s + (−0.654 − 0.377i)7-s + (0.744 − 1.28i)9-s + (0.522 + 0.904i)11-s + (−0.997 − 0.0643i)13-s + (−1.05 + 0.610i)15-s + (0.0562 − 0.0974i)17-s + (−0.542 + 0.940i)19-s + 1.19·21-s + (−0.493 − 0.854i)23-s − 0.400·25-s + 0.769i·27-s + (−0.0746 + 0.0430i)29-s − 0.0352i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.759 + 0.650i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.759 + 0.650i$
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: \(1\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.759 + 0.650i)\)

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 \( 1 \)
13 \( 1 + (3.59 + 0.232i)T \)
good3 \( 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2} \)
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 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.36 - 4.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.401 - 0.232i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.196iT - 31T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (10.7 + 6.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.6iT - 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 + (-0.464 + 0.803i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.5 + 6.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.09 + 7.09i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.0 - 5.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.19iT - 73T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + (13.3 - 7.73i)T + (44.5 - 77.0i)T^{2} \)
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

−9.937320817230877243006730226984, −9.693092048785965799108138466402, −8.238864718995406736665597526339, −6.80090567775694803637690421120, −6.43255493432673088387957864134, −5.37965047500399563365257126725, −4.66932085531096349826805844152, −3.68258941768307417564459961712, −1.95926020103333398719218504371, 0, 1.55127966613772653635832194470, 2.86595080290625693495463666517, 4.52477829635618333634785034522, 5.72123882800760642411037093895, 6.01685940753180579280486418089, 6.83974332914137070798219456170, 7.72408930626670411408262597742, 9.075602396640237623680029351479, 9.700766675820996941323991569886

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