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Properties

Label 2-847-11.5-c1-0-48
Degree $2$
Conductor $847$
Sign $-0.598 + 0.801i$
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  + (−0.254 − 0.783i)2-s + (0.777 + 0.564i)3-s + (1.06 − 0.776i)4-s + (0.922 − 2.83i)5-s + (0.244 − 0.752i)6-s + (−0.809 + 0.587i)7-s + (−2.21 − 1.60i)8-s + (−0.641 − 1.97i)9-s − 2.45·10-s + 1.26·12-s + (0.680 + 2.09i)13-s + (0.666 + 0.484i)14-s + (2.32 − 1.68i)15-s + (0.120 − 0.371i)16-s + (1.36 − 4.19i)17-s + (−1.38 + 1.00i)18-s + ⋯
L(s)  = 1  + (−0.179 − 0.553i)2-s + (0.448 + 0.326i)3-s + (0.534 − 0.388i)4-s + (0.412 − 1.26i)5-s + (0.0998 − 0.307i)6-s + (−0.305 + 0.222i)7-s + (−0.782 − 0.568i)8-s + (−0.213 − 0.658i)9-s − 0.777·10-s + 0.366·12-s + (0.188 + 0.580i)13-s + (0.178 + 0.129i)14-s + (0.599 − 0.435i)15-s + (0.0301 − 0.0928i)16-s + (0.330 − 1.01i)17-s + (−0.326 + 0.236i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.598 + 0.801i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.598 + 0.801i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794050 - 1.58389i\)
\(L(\frac12)\) \(\approx\) \(0.794050 - 1.58389i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.254 + 0.783i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.777 - 0.564i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.922 + 2.83i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.680 - 2.09i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.36 + 4.19i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.39 - 1.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 8.39T + 23T^{2} \)
29 \( 1 + (2.66 - 1.93i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.31 - 7.10i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.10 + 5.16i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.36 + 3.17i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 9.44T + 43T^{2} \)
47 \( 1 + (-4.36 - 3.17i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.90 - 8.93i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.81 - 2.04i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.01 + 12.3i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 4.32T + 67T^{2} \)
71 \( 1 + (-1.35 + 4.18i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (11.9 - 8.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.22 + 6.83i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.35 + 7.26i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (-0.882 - 2.71i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.638430224582570342290763745021, −9.306948170597542465795656523389, −8.631627337400888235182424478989, −7.39014355604498200967542863351, −6.19126622972868615658453149761, −5.57899975360575810557857753402, −4.33619407765819135446914321872, −3.23608159476434790381657588250, −2.08566897591914064557610547994, −0.841916880881658289615432611744, 2.14060521295334176699812399180, 2.83606135399464361980097654956, 3.86953773890988552330607682975, 5.76656277656968578817091118017, 6.21264103515006181638918873117, 7.19017999774267776227450860755, 7.85410318768894060097922950391, 8.378079653430656186872897895021, 9.746426934219694972356382579547, 10.43921871186671410990036015162

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