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Properties

Label 2-832-13.12-c1-0-24
Degree $2$
Conductor $832$
Sign $-0.554 + 0.832i$
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  + 3-s − 3i·5-s − 3i·7-s − 2·9-s + (−2 + 3i)13-s − 3i·15-s + 3·17-s − 6i·19-s − 3i·21-s − 6·23-s − 4·25-s − 5·27-s − 9·35-s + 3i·37-s + (−2 + 3i)39-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34i·5-s − 1.13i·7-s − 0.666·9-s + (−0.554 + 0.832i)13-s − 0.774i·15-s + 0.727·17-s − 1.37i·19-s − 0.654i·21-s − 1.25·23-s − 0.800·25-s − 0.962·27-s − 1.52·35-s + 0.493i·37-s + (−0.320 + 0.480i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.663969 - 1.24063i\)
\(L(\frac12)\) \(\approx\) \(0.663969 - 1.24063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3i)T \)
good3 \( 1 - T + 3T^{2} \)
5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + 15iT - 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.651645694321059600876178493387, −9.119591276734565632001334280508, −8.252038045666852367944499448759, −7.58617998981573004755675415268, −6.53436024154874445688464659371, −5.26244449010756976081609202603, −4.49573098618968946685002340039, −3.55132452797694666876388426541, −2.09692158906131207649188068075, −0.61664620201749696088752065019, 2.23070782087878359389398155470, 2.90400549729722709434722307779, 3.79289041969241534411196168571, 5.62829483129110191656787325037, 5.90077297198346321050112302243, 7.22856110240818040973853763575, 8.010318375320170242315563239358, 8.673263067072773529570791574197, 9.854317223805585221182738574299, 10.27159169386435966546163094468

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