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Properties

Label 2-847-11.3-c1-0-7
Degree $2$
Conductor $847$
Sign $-0.995 - 0.0913i$
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  + (−1.96 + 1.42i)2-s + (0.443 + 1.36i)3-s + (1.20 − 3.71i)4-s + (−1.01 − 0.739i)5-s + (−2.82 − 2.04i)6-s + (−0.309 + 0.951i)7-s + (1.43 + 4.40i)8-s + (0.762 − 0.554i)9-s + 3.05·10-s + 5.60·12-s + (−2.57 + 1.86i)13-s + (−0.751 − 2.31i)14-s + (0.557 − 1.71i)15-s + (−2.79 − 2.02i)16-s + (4.79 + 3.48i)17-s + (−0.708 + 2.17i)18-s + ⋯
L(s)  = 1  + (−1.39 + 1.01i)2-s + (0.255 + 0.787i)3-s + (0.603 − 1.85i)4-s + (−0.455 − 0.330i)5-s + (−1.15 − 0.836i)6-s + (−0.116 + 0.359i)7-s + (0.506 + 1.55i)8-s + (0.254 − 0.184i)9-s + 0.967·10-s + 1.61·12-s + (−0.713 + 0.518i)13-s + (−0.200 − 0.617i)14-s + (0.144 − 0.443i)15-s + (−0.698 − 0.507i)16-s + (1.16 + 0.844i)17-s + (−0.166 + 0.513i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.995 - 0.0913i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.995 - 0.0913i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0271821 + 0.593606i\)
\(L(\frac12)\) \(\approx\) \(0.0271821 + 0.593606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.96 - 1.42i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.443 - 1.36i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.01 + 0.739i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.57 - 1.86i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.79 - 3.48i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.884 + 2.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.76T + 23T^{2} \)
29 \( 1 + (1.38 - 4.27i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.86 - 5.71i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.68 - 5.18i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.0971 - 0.299i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.132T + 43T^{2} \)
47 \( 1 + (-2.89 - 8.91i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.52 - 2.55i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.14 - 6.60i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.98 + 1.44i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.41T + 67T^{2} \)
71 \( 1 + (-0.0943 - 0.0685i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.190 - 0.584i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.90 + 5.01i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.768 - 0.558i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (14.2 - 10.3i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.34489262140254209605234445458, −9.226903894812093892499484253374, −9.180195950289035955919545174714, −8.197733872360062282224445760556, −7.32925369459026654912424304106, −6.62766173016484804333343182242, −5.46408857408005906752796339198, −4.54834502154124112939382401689, −3.23708843661694447428717887362, −1.34060317763591232341394295975, 0.49217316858746146802944468552, 1.75670325271314239649156372400, 2.81965789021350875164653805675, 3.74102849367853940368858424578, 5.37251226939113406393850494970, 7.02157738395309157844046700324, 7.56941395553195321316153344322, 7.924299349120777245151660366213, 9.119846991575198785992365005356, 9.799224960152610759777964101229

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