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Properties

Label 2-847-11.3-c1-0-38
Degree $2$
Conductor $847$
Sign $0.0694 + 0.997i$
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  + (1.14 − 0.828i)2-s + (−0.668 − 2.05i)3-s + (−0.00431 + 0.0132i)4-s + (1.48 + 1.07i)5-s + (−2.46 − 1.79i)6-s + (0.309 − 0.951i)7-s + (0.877 + 2.69i)8-s + (−1.35 + 0.987i)9-s + 2.58·10-s + 0.0302·12-s + (3.75 − 2.73i)13-s + (−0.435 − 1.34i)14-s + (1.22 − 3.76i)15-s + (3.21 + 2.33i)16-s + (4.42 + 3.21i)17-s + (−0.731 + 2.25i)18-s + ⋯
L(s)  = 1  + (0.806 − 0.585i)2-s + (−0.385 − 1.18i)3-s + (−0.00215 + 0.00664i)4-s + (0.662 + 0.481i)5-s + (−1.00 − 0.731i)6-s + (0.116 − 0.359i)7-s + (0.310 + 0.954i)8-s + (−0.453 + 0.329i)9-s + 0.816·10-s + 0.00872·12-s + (1.04 − 0.757i)13-s + (−0.116 − 0.358i)14-s + (0.316 − 0.972i)15-s + (0.803 + 0.583i)16-s + (1.07 + 0.779i)17-s + (−0.172 + 0.530i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0694 + 0.997i$
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: \((2,\ 847,\ (\ :1/2),\ 0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80188 - 1.68075i\)
\(L(\frac12)\) \(\approx\) \(1.80188 - 1.68075i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 0.828i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.668 + 2.05i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.48 - 1.07i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-3.75 + 2.73i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.42 - 3.21i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.79 + 5.51i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.719T + 23T^{2} \)
29 \( 1 + (0.362 - 1.11i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.05 + 0.768i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.647 + 1.99i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.283 + 0.871i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 8.02T + 43T^{2} \)
47 \( 1 + (-1.84 - 5.68i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (8.20 - 5.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.37 - 7.30i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.07 - 3.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + (11.2 + 8.17i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.85 + 5.72i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-12.6 + 9.19i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.54 - 2.57i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (1.95 - 1.41i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.58718615141222945575681916959, −9.118248322763946619085922524570, −8.004761609839234322126233886676, −7.43241891480678286186674192798, −6.20007985476724368254407651862, −5.83488020028073242694566692820, −4.50476131066313133017573897349, −3.36920757068237324538182903303, −2.34853199084089688051613816162, −1.19272142416508648502855388190, 1.51818545247416473353532846189, 3.53146368505313808203494584749, 4.30500485076058427060586175438, 5.21613263246442145441019229683, 5.70692073192049122573241247648, 6.46308108958923671200669599276, 7.77026255077264151449982698667, 8.973446455509300739032632147694, 9.669707259412415573999366421819, 10.19309153221726724328902549869

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