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Properties

Label 2-847-11.9-c1-0-42
Degree $2$
Conductor $847$
Sign $-0.353 + 0.935i$
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  + (0.453 − 1.39i)2-s + (1.30 − 0.951i)3-s + (−0.122 − 0.0889i)4-s + (0.144 + 0.443i)5-s + (−0.733 − 2.25i)6-s + (−0.809 − 0.587i)7-s + (2.19 − 1.59i)8-s + (−0.118 + 0.363i)9-s + 0.684·10-s − 0.244·12-s + (0.488 − 1.50i)13-s + (−1.18 + 0.862i)14-s + (0.610 + 0.443i)15-s + (−1.32 − 4.07i)16-s + (−1.61 − 4.97i)17-s + (0.453 + 0.329i)18-s + ⋯
L(s)  = 1  + (0.320 − 0.986i)2-s + (0.755 − 0.549i)3-s + (−0.0612 − 0.0444i)4-s + (0.0645 + 0.198i)5-s + (−0.299 − 0.921i)6-s + (−0.305 − 0.222i)7-s + (0.775 − 0.563i)8-s + (−0.0393 + 0.121i)9-s + 0.216·10-s − 0.0706·12-s + (0.135 − 0.417i)13-s + (−0.317 + 0.230i)14-s + (0.157 + 0.114i)15-s + (−0.330 − 1.01i)16-s + (−0.391 − 1.20i)17-s + (0.106 + 0.0776i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.353 + 0.935i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47672 - 2.13761i\)
\(L(\frac12)\) \(\approx\) \(1.47672 - 2.13761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.453 + 1.39i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.144 - 0.443i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.488 + 1.50i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.61 + 4.97i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.41 + 2.48i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.80T + 23T^{2} \)
29 \( 1 + (-2.20 - 1.59i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.399 + 1.23i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.57 - 1.14i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.842 - 0.611i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.70T + 43T^{2} \)
47 \( 1 + (-5.17 + 3.75i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (4.08 - 12.5i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.96 - 5.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.70 - 14.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.67T + 67T^{2} \)
71 \( 1 + (-3.01 - 9.26i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (10.7 + 7.82i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.10 - 3.40i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.33 - 16.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.91T + 89T^{2} \)
97 \( 1 + (0.835 - 2.57i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.12557453350060067137009685867, −9.194927265346821907603883841270, −8.237467290090300047843344119136, −7.29316567353475753733221203426, −6.81442269884696918748070066294, −5.28542442118511061854063722901, −4.19404222318733913830163823094, −2.89864322195123641820785667753, −2.64551490635882358973145110195, −1.16157412672310845120196099841, 1.82402724186866923253783486427, 3.26700311448267674788833717789, 4.21971745057170880020744584004, 5.24599307414804197832367523883, 6.19347577790718572713262161255, 6.83822395944753787702743691710, 8.043503048485559272718285383521, 8.546499987558621772246707959893, 9.486683038677446544912482246577, 10.22156778215849837145606206813

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