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Properties

Label 2-847-11.5-c1-0-52
Degree $2$
Conductor $847$
Sign $-0.957 - 0.288i$
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  + (−0.421 − 1.29i)2-s + (−0.293 − 0.213i)3-s + (0.114 − 0.0830i)4-s + (0.970 − 2.98i)5-s + (−0.153 + 0.471i)6-s + (0.809 − 0.587i)7-s + (−2.36 − 1.71i)8-s + (−0.886 − 2.72i)9-s − 4.28·10-s − 0.0513·12-s + (−1.47 − 4.54i)13-s + (−1.10 − 0.801i)14-s + (−0.923 + 0.670i)15-s + (−1.14 + 3.51i)16-s + (−1.47 + 4.54i)17-s + (−3.16 + 2.29i)18-s + ⋯
L(s)  = 1  + (−0.297 − 0.916i)2-s + (−0.169 − 0.123i)3-s + (0.0571 − 0.0415i)4-s + (0.434 − 1.33i)5-s + (−0.0624 + 0.192i)6-s + (0.305 − 0.222i)7-s + (−0.835 − 0.606i)8-s + (−0.295 − 0.909i)9-s − 1.35·10-s − 0.0148·12-s + (−0.409 − 1.26i)13-s + (−0.294 − 0.214i)14-s + (−0.238 + 0.173i)15-s + (−0.285 + 0.879i)16-s + (−0.358 + 1.10i)17-s + (−0.745 + 0.541i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.957 - 0.288i$
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: \((2,\ 847,\ (\ :1/2),\ -0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.191544 + 1.30182i\)
\(L(\frac12)\) \(\approx\) \(0.191544 + 1.30182i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.421 + 1.29i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (0.293 + 0.213i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.970 + 2.98i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.47 + 4.54i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.47 - 4.54i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.67 - 4.11i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.14T + 23T^{2} \)
29 \( 1 + (5.67 - 4.11i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.12 + 3.45i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-7.98 + 5.80i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.60 - 1.89i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 4.28T + 43T^{2} \)
47 \( 1 + (-0.629 - 0.457i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.705 - 2.17i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.293 + 0.213i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.995 + 3.06i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.59T + 67T^{2} \)
71 \( 1 + (4.68 - 14.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.60 + 1.89i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.14 + 3.53i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.480 + 1.47i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 + (-1.90 - 5.84i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.618607288955109521968990609421, −9.242969453649278507712877556957, −8.288131687007146085339415104047, −7.26741906634015573798595281332, −5.84099528732153322373393638859, −5.53436654894266100491781199685, −4.08190976673990535625067187779, −3.00095527361905711777154652304, −1.54165459308531058857531502027, −0.73905106640712407794823579834, 2.33123621712513767621339228602, 2.93682120733680784153308237835, 4.74781341692426229733387075573, 5.58230373195453919698301960975, 6.58755704539296320078445446214, 7.20640816089347836569711360287, 7.72562721391323968249656915099, 9.050677333061690322389672662284, 9.528419325990580082169235341595, 10.78371347645792464549524435804

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