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Properties

Label 2-832-52.31-c1-0-21
Degree $2$
Conductor $832$
Sign $-0.916 + 0.399i$
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 + (−2.16 + 2.16i)11-s + (−0.418 − 3.58i)13-s + (1.58 + 1.58i)15-s − 5.32i·17-s + (−5.16 − 5.16i)19-s + (1.58 + 1.58i)21-s − 0.837·23-s − 5i·27-s − 5.16·29-s + (−5.16 − 5.16i)31-s + (2.16 + 2.16i)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 + (−0.651 + 0.651i)11-s + (−0.116 − 0.993i)13-s + (0.408 + 0.408i)15-s − 1.29i·17-s + (−1.18 − 1.18i)19-s + (0.345 + 0.345i)21-s − 0.174·23-s − 0.962i·27-s − 0.958·29-s + (−0.927 − 0.927i)31-s + (0.376 + 0.376i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 + 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $-0.916 + 0.399i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ -0.916 + 0.399i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0644345 - 0.309561i\)
\(L(\frac12)\) \(\approx\) \(0.0644345 - 0.309561i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.418 + 3.58i)T \)
good3 \( 1 + iT - 3T^{2} \)
5 \( 1 + (1.58 - 1.58i)T - 5iT^{2} \)
7 \( 1 + (1.58 - 1.58i)T - 7iT^{2} \)
11 \( 1 + (2.16 - 2.16i)T - 11iT^{2} \)
17 \( 1 + 5.32iT - 17T^{2} \)
19 \( 1 + (5.16 + 5.16i)T + 19iT^{2} \)
23 \( 1 + 0.837T + 23T^{2} \)
29 \( 1 + 5.16T + 29T^{2} \)
31 \( 1 + (5.16 + 5.16i)T + 31iT^{2} \)
37 \( 1 + (-0.418 - 0.418i)T + 37iT^{2} \)
41 \( 1 + (1.16 - 1.16i)T - 41iT^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (2.74 - 2.74i)T - 47iT^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + (4 - 4i)T - 59iT^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + (-5.32 - 5.32i)T + 67iT^{2} \)
71 \( 1 + (1.58 + 1.58i)T + 71iT^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 15.4iT - 79T^{2} \)
83 \( 1 + (-12.1 - 12.1i)T + 83iT^{2} \)
89 \( 1 + (9.16 + 9.16i)T + 89iT^{2} \)
97 \( 1 + (10.1 - 10.1i)T - 97iT^{2} \)
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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

−9.867372595472121538325236328496, −9.052569281431609077859926535143, −7.81172137108504189357303348753, −7.32074279751196202585049449167, −6.62174779970296582739392750507, −5.48019395920802573787781343012, −4.37983264279784316451336793887, −3.09428415314027232201116418594, −2.25288487243295406711082711960, −0.14844604267786918251094262128, 1.73133344228739641067116608804, 3.78628784388494463913476118255, 3.93589391652996748306899650805, 5.09947955730652492697323486017, 6.24982682638733796889779509273, 7.19324068356944653037168317602, 8.213806234456429725807504407086, 8.797963965303228813876269835145, 9.898511690464895932606735839967, 10.46465667009714537382434523075

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