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Properties

Label 2-847-11.3-c1-0-52
Degree $2$
Conductor $847$
Sign $-0.751 - 0.659i$
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  + (1.86 − 1.35i)2-s + (−0.711 − 2.19i)3-s + (1.02 − 3.14i)4-s + (−2.91 − 2.11i)5-s + (−4.29 − 3.11i)6-s + (0.309 − 0.951i)7-s + (−0.927 − 2.85i)8-s + (−1.86 + 1.35i)9-s − 8.30·10-s − 7.60·12-s + (5.34 − 3.88i)13-s + (−0.711 − 2.19i)14-s + (−2.56 + 7.89i)15-s + (−0.244 − 0.177i)16-s + (2.18 + 1.58i)17-s + (−1.63 + 5.04i)18-s + ⋯
L(s)  = 1  + (1.31 − 0.957i)2-s + (−0.410 − 1.26i)3-s + (0.510 − 1.57i)4-s + (−1.30 − 0.947i)5-s + (−1.75 − 1.27i)6-s + (0.116 − 0.359i)7-s + (−0.327 − 1.00i)8-s + (−0.620 + 0.451i)9-s − 2.62·10-s − 2.19·12-s + (1.48 − 1.07i)13-s + (−0.190 − 0.585i)14-s + (−0.662 + 2.03i)15-s + (−0.0612 − 0.0444i)16-s + (0.529 + 0.384i)17-s + (−0.386 + 1.18i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.751 - 0.659i$
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: \((2,\ 847,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783744 + 2.08270i\)
\(L(\frac12)\) \(\approx\) \(0.783744 + 2.08270i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.86 + 1.35i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.711 + 2.19i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.91 + 2.11i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-5.34 + 3.88i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.18 - 1.58i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.927 - 2.85i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.69T + 23T^{2} \)
29 \( 1 + (1.45 - 4.46i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.61 - 4.95i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.16 + 6.65i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.69T + 43T^{2} \)
47 \( 1 + (0.589 + 1.81i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-10.4 + 7.58i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.06 + 6.36i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.48 + 2.52i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.51T + 67T^{2} \)
71 \( 1 + (-3.48 - 2.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.54 + 4.75i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.71 - 4.88i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.42 - 1.76i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 + (-2.91 + 2.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.14952338095798724642479296110, −8.369270489747793418173487093986, −8.076216350857630390972502712087, −6.95180813301497262778806711343, −5.82676265358446095606901834465, −5.18772780107366856675512642190, −3.90781908218206452195894381778, −3.46789675293851133003824933939, −1.64989965660975550792934066955, −0.816540821389880715799656404903, 3.02879663177000195087014911692, 3.99759095526103114674106524841, 4.20745020378437958043903235560, 5.36713580601807496407402169075, 6.19826646043043787302508205690, 7.04572370866402484160391973025, 7.85206525025858385184621915627, 8.885101214484670115070846858340, 10.00789063059091523380808335745, 11.06064173702522927682412342959

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