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Properties

Label 2-847-77.19-c1-0-22
Degree $2$
Conductor $847$
Sign $0.975 + 0.220i$
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  + (−0.0270 + 0.00284i)2-s + (0.914 + 0.823i)3-s + (−1.95 + 0.415i)4-s + (−0.470 − 1.05i)5-s + (−0.0270 − 0.0196i)6-s + (−2.40 + 1.09i)7-s + (0.103 − 0.0336i)8-s + (−0.155 − 1.47i)9-s + (0.0157 + 0.0272i)10-s + (−2.13 − 1.23i)12-s + (4.64 − 3.37i)13-s + (0.0620 − 0.0364i)14-s + (0.440 − 1.35i)15-s + (3.65 − 1.62i)16-s + (−0.652 + 6.21i)17-s + (0.00839 + 0.0394i)18-s + ⋯
L(s)  = 1  + (−0.0191 + 0.00200i)2-s + (0.528 + 0.475i)3-s + (−0.977 + 0.207i)4-s + (−0.210 − 0.472i)5-s + (−0.0110 − 0.00803i)6-s + (−0.910 + 0.413i)7-s + (0.0365 − 0.0118i)8-s + (−0.0517 − 0.492i)9-s + (0.00497 + 0.00861i)10-s + (−0.615 − 0.355i)12-s + (1.28 − 0.935i)13-s + (0.0165 − 0.00973i)14-s + (0.113 − 0.349i)15-s + (0.912 − 0.406i)16-s + (−0.158 + 1.50i)17-s + (0.00197 + 0.00930i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28258 - 0.143040i\)
\(L(\frac12)\) \(\approx\) \(1.28258 - 0.143040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.40 - 1.09i)T \)
11 \( 1 \)
good2 \( 1 + (0.0270 - 0.00284i)T + (1.95 - 0.415i)T^{2} \)
3 \( 1 + (-0.914 - 0.823i)T + (0.313 + 2.98i)T^{2} \)
5 \( 1 + (0.470 + 1.05i)T + (-3.34 + 3.71i)T^{2} \)
13 \( 1 + (-4.64 + 3.37i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.652 - 6.21i)T + (-16.6 - 3.53i)T^{2} \)
19 \( 1 + (-2.31 - 0.492i)T + (17.3 + 7.72i)T^{2} \)
23 \( 1 + (-2.75 + 4.77i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.27 - 1.39i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.809 - 1.81i)T + (-20.7 - 23.0i)T^{2} \)
37 \( 1 + (-1.85 - 2.06i)T + (-3.86 + 36.7i)T^{2} \)
41 \( 1 + (0.435 + 1.34i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.93iT - 43T^{2} \)
47 \( 1 + (-0.969 + 4.55i)T + (-42.9 - 19.1i)T^{2} \)
53 \( 1 + (-8.16 - 3.63i)T + (35.4 + 39.3i)T^{2} \)
59 \( 1 + (1.83 + 8.64i)T + (-53.8 + 23.9i)T^{2} \)
61 \( 1 + (1.82 - 0.810i)T + (40.8 - 45.3i)T^{2} \)
67 \( 1 + (5.00 + 8.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.374 + 0.272i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.295 - 0.0627i)T + (66.6 - 29.6i)T^{2} \)
79 \( 1 + (6.40 - 0.673i)T + (77.2 - 16.4i)T^{2} \)
83 \( 1 + (3.29 + 2.39i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-12.9 - 7.49i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.88 + 3.96i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.08894758480147488448430186878, −9.049636143944236880147146252439, −8.648477354876583084560127110497, −8.120576881239479868375830763191, −6.53593813008159787741912024724, −5.75627705660112162943805505725, −4.57910733042881827110099253256, −3.67124543238247096065297352882, −3.06121513607888139339963743227, −0.803367717870539306769671281867, 1.10963143563693613305977290719, 2.83707221175329541361220856980, 3.70061994488681789137816863662, 4.77959059344704901105764050334, 5.89640109947788419917353013742, 7.02370516156340341507826076156, 7.53229333031741055842425585340, 8.763495586747754713132802893791, 9.191656843856781754010173756780, 10.07706286111999700727173248912

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