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Properties

Label 2-832-52.47-c1-0-9
Degree $2$
Conductor $832$
Sign $0.176 - 0.984i$
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  + i·3-s + (1.58 + 1.58i)5-s + (1.58 + 1.58i)7-s + 2·9-s + (4.16 + 4.16i)11-s + (−3.58 + 0.418i)13-s + (−1.58 + 1.58i)15-s − 7.32i·17-s + (1.16 − 1.16i)19-s + (−1.58 + 1.58i)21-s − 7.16·23-s + 5i·27-s + 1.16·29-s + (1.16 − 1.16i)31-s + (−4.16 + 4.16i)33-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.707 + 0.707i)5-s + (0.597 + 0.597i)7-s + 0.666·9-s + (1.25 + 1.25i)11-s + (−0.993 + 0.116i)13-s + (−0.408 + 0.408i)15-s − 1.77i·17-s + (0.266 − 0.266i)19-s + (−0.345 + 0.345i)21-s − 1.49·23-s + 0.962i·27-s + 0.215·29-s + (0.208 − 0.208i)31-s + (−0.724 + 0.724i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.176 - 0.984i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.176 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53429 + 1.28346i\)
\(L(\frac12)\) \(\approx\) \(1.53429 + 1.28346i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.58 - 0.418i)T \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 + (-1.58 - 1.58i)T + 5iT^{2} \)
7 \( 1 + (-1.58 - 1.58i)T + 7iT^{2} \)
11 \( 1 + (-4.16 - 4.16i)T + 11iT^{2} \)
17 \( 1 + 7.32iT - 17T^{2} \)
19 \( 1 + (-1.16 + 1.16i)T - 19iT^{2} \)
23 \( 1 + 7.16T + 23T^{2} \)
29 \( 1 - 1.16T + 29T^{2} \)
31 \( 1 + (-1.16 + 1.16i)T - 31iT^{2} \)
37 \( 1 + (-3.58 + 3.58i)T - 37iT^{2} \)
41 \( 1 + (-5.16 - 5.16i)T + 41iT^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-6.74 - 6.74i)T + 47iT^{2} \)
53 \( 1 + 9.48T + 53T^{2} \)
59 \( 1 + (4 + 4i)T + 59iT^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (7.32 - 7.32i)T - 67iT^{2} \)
71 \( 1 + (-1.58 + 1.58i)T - 71iT^{2} \)
73 \( 1 + (6 - 6i)T - 73iT^{2} \)
79 \( 1 + 3.48iT - 79T^{2} \)
83 \( 1 + (-5.83 + 5.83i)T - 83iT^{2} \)
89 \( 1 + (2.83 - 2.83i)T - 89iT^{2} \)
97 \( 1 + (3.83 + 3.83i)T + 97iT^{2} \)
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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

−10.00628198435242968847048871249, −9.730510680209070965464954466179, −9.124698414770746559910754034372, −7.62717068919599713478781513719, −7.01570927895490141020769344955, −6.06415238159019724414418488628, −4.83577766874624744071703040557, −4.33049498478028715275291539062, −2.73850132333793788318946288493, −1.80589258387814198434170554441, 1.13055972747717683391360899055, 1.87662111953807492038533167085, 3.72861615547987754397050507290, 4.56478013553522972578217505363, 5.83980890013127581453489487632, 6.40678641852645980965417642592, 7.58676259482904407910120601333, 8.247893994105642261646282968000, 9.151282269567431503194652708632, 10.03612511894147779435084905340

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