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Properties

Label 2-847-11.4-c1-0-42
Degree $2$
Conductor $847$
Sign $0.642 - 0.766i$
Analytic cond. $6.76332$
Root an. cond. $2.60064$
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  + (2.23 + 1.62i)2-s + (0.544 − 1.67i)3-s + (1.73 + 5.35i)4-s + (2.12 − 1.54i)5-s + (3.93 − 2.85i)6-s + (−0.309 − 0.951i)7-s + (−3.09 + 9.52i)8-s + (−0.0833 − 0.0605i)9-s + 7.25·10-s + 9.91·12-s + (−1.93 − 1.40i)13-s + (0.853 − 2.62i)14-s + (−1.42 − 4.39i)15-s + (−13.2 + 9.64i)16-s + (−1.93 + 1.40i)17-s + (−0.0879 − 0.270i)18-s + ⋯
L(s)  = 1  + (1.57 + 1.14i)2-s + (0.314 − 0.967i)3-s + (0.869 + 2.67i)4-s + (0.950 − 0.690i)5-s + (1.60 − 1.16i)6-s + (−0.116 − 0.359i)7-s + (−1.09 + 3.36i)8-s + (−0.0277 − 0.0201i)9-s + 2.29·10-s + 2.86·12-s + (−0.535 − 0.389i)13-s + (0.228 − 0.701i)14-s + (−0.369 − 1.13i)15-s + (−3.31 + 2.41i)16-s + (−0.468 + 0.340i)17-s + (−0.0207 − 0.0638i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.13449 + 1.92865i\)
\(L(\frac12)\) \(\approx\) \(4.13449 + 1.92865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-2.23 - 1.62i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.544 + 1.67i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.12 + 1.54i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (1.93 + 1.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.93 - 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.534 + 1.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.626T + 23T^{2} \)
29 \( 1 + (0.534 + 1.64i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.81 - 1.31i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.13 + 6.55i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.20 - 9.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 7.25T + 43T^{2} \)
47 \( 1 + (-1.97 + 6.07i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.48 - 5.43i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.544 + 1.67i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (8.40 - 6.10i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.42T + 67T^{2} \)
71 \( 1 + (6.54 - 4.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.20 + 9.87i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.3 - 8.96i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.3 + 7.50i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-6.75 - 4.91i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.35895552139532424710073879584, −9.041509694586959481447319134569, −8.191976467932221738282991516735, −7.43151125260362346369519071576, −6.74576363292872223494431309936, −5.96297738832091449921465896535, −5.11449064223870203798300330688, −4.31426171099306551144981699947, −2.95609300377199411925271132969, −1.89209099181309468516671159340, 1.83525396397204908301591983170, 2.75176517843696163225953596956, 3.55891596654382744025298522774, 4.55329307496542521459380384261, 5.28626084411492727604292622771, 6.23637096425349663080537334065, 6.94929525839162309324125023329, 8.995987401804115684521138913412, 9.758873428111936105071041342608, 10.20706782636151422680163133819

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