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Properties

Label 2-832-13.10-c1-0-17
Degree $2$
Conductor $832$
Sign $-0.252 + 0.967i$
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  + (−0.5 − 0.866i)3-s + (−1.5 − 0.866i)7-s + (1 − 1.73i)9-s + (4.5 − 2.59i)11-s + (1 + 3.46i)13-s + (−1.5 + 2.59i)17-s + (−4.5 − 2.59i)19-s + 1.73i·21-s + (−1.5 − 2.59i)23-s + 5·25-s − 5·27-s + (−4.5 − 7.79i)29-s − 3.46i·31-s + (−4.5 − 2.59i)33-s + (4.5 − 2.59i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.566 − 0.327i)7-s + (0.333 − 0.577i)9-s + (1.35 − 0.783i)11-s + (0.277 + 0.960i)13-s + (−0.363 + 0.630i)17-s + (−1.03 − 0.596i)19-s + 0.377i·21-s + (−0.312 − 0.541i)23-s + 25-s − 0.962·27-s + (−0.835 − 1.44i)29-s − 0.622i·31-s + (−0.783 − 0.452i)33-s + (0.739 − 0.427i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741852 - 0.960409i\)
\(L(\frac12)\) \(\approx\) \(0.741852 - 0.960409i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 3.46i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.5 + 2.59i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.3iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (4.5 + 2.59i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.5 - 2.59i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-13.5 + 7.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.874536189699665475040117024040, −9.086791834372056325524199901401, −8.433458922398406393901620027393, −7.03247710124247652629867331484, −6.52792097180569563118453043224, −5.97805564991413133882543913643, −4.23721928055528600834165874321, −3.74849231747684227131137224695, −2.05014918433209941702682463525, −0.64423986430193114409053835610, 1.61662404485834749268950047877, 3.12144522065742916507120424546, 4.20729779576534676034351976271, 5.05533891573045106781388617517, 6.10619643422359680120666426981, 6.92711988312413531601389032534, 7.893226401697387894349499189095, 9.046231007779870136382547417773, 9.555505742437537801958592866682, 10.53300007832769016972240233361

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