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Properties

Label 2-832-8.5-c1-0-14
Degree $2$
Conductor $832$
Sign $0.707 + 0.707i$
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  + 0.414i·3-s − 3.82i·5-s + 1.58·7-s + 2.82·9-s + 4.82i·11-s + i·13-s + 1.58·15-s + 17-s − 5.65i·19-s + 0.656i·21-s + 3.17·23-s − 9.65·25-s + 2.41i·27-s − 7.65i·29-s + 7.65·31-s + ⋯
L(s)  = 1  + 0.239i·3-s − 1.71i·5-s + 0.599·7-s + 0.942·9-s + 1.45i·11-s + 0.277i·13-s + 0.409·15-s + 0.242·17-s − 1.29i·19-s + 0.143i·21-s + 0.661·23-s − 1.93·25-s + 0.464i·27-s − 1.42i·29-s + 1.37·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64449 - 0.681170i\)
\(L(\frac12)\) \(\approx\) \(1.64449 - 0.681170i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - 0.414iT - 3T^{2} \)
5 \( 1 + 3.82iT - 5T^{2} \)
7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
29 \( 1 + 7.65iT - 29T^{2} \)
31 \( 1 - 7.65T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 1.65T + 41T^{2} \)
43 \( 1 + 5.58iT - 43T^{2} \)
47 \( 1 + 9.24T + 47T^{2} \)
53 \( 1 - 7.65iT - 53T^{2} \)
59 \( 1 - 8iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 3.17iT - 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 - 9.65T + 73T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 - 16.1iT - 83T^{2} \)
89 \( 1 + 2.34T + 89T^{2} \)
97 \( 1 - 3.65T + 97T^{2} \)
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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

−9.714004693987941119211912804985, −9.474240753983576280240553769871, −8.469706518079776698292551020068, −7.66021114897178668741216946346, −6.77293698087494270644887960098, −5.31558943302594466017904167780, −4.60549947786170833939981831188, −4.23397165496940752623373231852, −2.16188283112582154733223292689, −1.04407452574058453280408384602, 1.45888006183022756313326972294, 2.95145589285992764616782277448, 3.61556779580791298077273297534, 5.05251101358870115694250175479, 6.26887304899866316696005646534, 6.73614122161069076279188744652, 7.81701851002899209713460732231, 8.302217867562039739066971583252, 9.785304635584572648354327627400, 10.37965835808982879777434371122

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