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Properties

Label 2-847-11.9-c1-0-34
Degree $2$
Conductor $847$
Sign $0.998 + 0.0475i$
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  + (0.118 − 0.363i)2-s + (1.30 − 0.951i)3-s + (1.5 + 1.08i)4-s + (0.309 + 0.951i)5-s + (−0.190 − 0.587i)6-s + (0.809 + 0.587i)7-s + (1.19 − 0.865i)8-s + (−0.118 + 0.363i)9-s + 0.381·10-s + 3·12-s + (0.381 − 1.17i)13-s + (0.309 − 0.224i)14-s + (1.30 + 0.951i)15-s + (0.972 + 2.99i)16-s + (−0.954 − 2.93i)17-s + (0.118 + 0.0857i)18-s + ⋯
L(s)  = 1  + (0.0834 − 0.256i)2-s + (0.755 − 0.549i)3-s + (0.750 + 0.544i)4-s + (0.138 + 0.425i)5-s + (−0.0779 − 0.239i)6-s + (0.305 + 0.222i)7-s + (0.421 − 0.305i)8-s + (−0.0393 + 0.121i)9-s + 0.120·10-s + 0.866·12-s + (0.105 − 0.326i)13-s + (0.0825 − 0.0600i)14-s + (0.337 + 0.245i)15-s + (0.243 + 0.747i)16-s + (−0.231 − 0.712i)17-s + (0.0278 + 0.0202i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.998 + 0.0475i$
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: \((2,\ 847,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.65345 - 0.0630948i\)
\(L(\frac12)\) \(\approx\) \(2.65345 - 0.0630948i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.118 + 0.363i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-0.381 + 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.954 + 2.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.42 - 1.03i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.09T + 23T^{2} \)
29 \( 1 + (3.73 + 2.71i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.30 - 4.02i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.23 + 3.80i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.04 - 6.57i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 12.5T + 43T^{2} \)
47 \( 1 + (-5.35 + 3.88i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (0.736 - 2.26i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.97 + 6.51i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.35 - 7.24i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + (4.97 + 15.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (11.5 + 8.36i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.97 - 6.06i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.836 + 2.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.85T + 89T^{2} \)
97 \( 1 + (-2.16 + 6.65i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.51397534862989022631015299233, −9.105524081106838101269968666194, −8.444082628051091609624165276744, −7.48259317343889643354000097420, −7.08920230744318530280918355196, −5.98418370102715604738170696132, −4.71509733835571152891468495353, −3.28985917305462466484186907242, −2.64558249657719489395238605924, −1.67863998118262186769247495039, 1.38509709871023067748507030764, 2.63492408511358114529748018156, 3.83046465835765592354759477273, 4.85486599999656808848809252067, 5.79874981825160988297900847431, 6.78719810983554850556011529541, 7.59558879020416234978928828372, 8.782729784139988014053399805585, 9.119748157475919916747506328322, 10.25926015887965380810159823532

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