Properties

Label 2-847-11.3-c1-0-15
Degree $2$
Conductor $847$
Sign $0.642 + 0.766i$
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  + (−1.71 + 1.24i)2-s + (−0.965 − 2.97i)3-s + (0.777 − 2.39i)4-s + (−0.392 − 0.284i)5-s + (5.37 + 3.90i)6-s + (−0.309 + 0.951i)7-s + (0.338 + 1.04i)8-s + (−5.47 + 3.97i)9-s + 1.03·10-s − 7.85·12-s + (4.53 − 3.29i)13-s + (−0.656 − 2.02i)14-s + (−0.468 + 1.44i)15-s + (2.18 + 1.58i)16-s + (4.53 + 3.29i)17-s + (4.44 − 13.6i)18-s + ⋯
L(s)  = 1  + (−1.21 + 0.883i)2-s + (−0.557 − 1.71i)3-s + (0.388 − 1.19i)4-s + (−0.175 − 0.127i)5-s + (2.19 + 1.59i)6-s + (−0.116 + 0.359i)7-s + (0.119 + 0.368i)8-s + (−1.82 + 1.32i)9-s + 0.325·10-s − 2.26·12-s + (1.25 − 0.914i)13-s + (−0.175 − 0.540i)14-s + (−0.120 + 0.372i)15-s + (0.546 + 0.397i)16-s + (1.10 + 0.799i)17-s + (1.04 − 3.22i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.642 + 0.766i$
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: \((2,\ 847,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.561460 - 0.261909i\)
\(L(\frac12)\) \(\approx\) \(0.561460 - 0.261909i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 - 1.24i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.965 + 2.97i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.392 + 0.284i)T + (1.54 + 4.75i)T^{2} \)
13 \( 1 + (-4.53 + 3.29i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.53 - 3.29i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.63 - 5.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.48T + 23T^{2} \)
29 \( 1 + (1.63 - 5.02i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.76 + 4.18i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.0726 - 0.223i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.738 + 2.27i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.03T + 43T^{2} \)
47 \( 1 + (0.497 + 1.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.45 + 1.78i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.965 + 2.97i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.93 + 1.40i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + (9.76 + 7.09i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.738 - 2.27i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.30 + 5.30i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (2.60 + 1.89i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 1.26T + 89T^{2} \)
97 \( 1 + (-7.11 + 5.16i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.01451572953398320463160064388, −8.706600672260812437529508640851, −8.104926552714253787275924598508, −7.75153463823265448640565412490, −6.73555553501055361907205622068, −5.99359537889816423962356818553, −5.58803366665771601276437771352, −3.37877310630726134653461839507, −1.63035730219212311065457509288, −0.71436147937900456493159645832, 0.944765638933769613081094012278, 2.93768474949354172817029935327, 3.71088469543886485365771477677, 4.77980457021841295471442099224, 5.75528042238464016704355926720, 7.02423200672302423168981950478, 8.262416967272333039881616590408, 9.146698089948223555070048112938, 9.540104446610915996406801323856, 10.25750194598412620130809064081

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