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Properties

Label 2-847-11.5-c1-0-5
Degree $2$
Conductor $847$
Sign $0.353 + 0.935i$
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  + (−0.613 − 1.88i)2-s + (−2.25 − 1.63i)3-s + (−1.56 + 1.13i)4-s + (−0.00832 + 0.0256i)5-s + (−1.70 + 5.25i)6-s + (0.809 − 0.587i)7-s + (−0.0990 − 0.0719i)8-s + (1.47 + 4.53i)9-s + 0.0534·10-s + 5.40·12-s + (1.50 + 4.64i)13-s + (−1.60 − 1.16i)14-s + (0.0606 − 0.0440i)15-s + (−1.27 + 3.91i)16-s + (−0.518 + 1.59i)17-s + (7.64 − 5.55i)18-s + ⋯
L(s)  = 1  + (−0.433 − 1.33i)2-s + (−1.30 − 0.945i)3-s + (−0.784 + 0.569i)4-s + (−0.00372 + 0.0114i)5-s + (−0.697 + 2.14i)6-s + (0.305 − 0.222i)7-s + (−0.0350 − 0.0254i)8-s + (0.490 + 1.51i)9-s + 0.0168·10-s + 1.55·12-s + (0.418 + 1.28i)13-s + (−0.429 − 0.311i)14-s + (0.0156 − 0.0113i)15-s + (−0.318 + 0.979i)16-s + (−0.125 + 0.387i)17-s + (1.80 − 1.30i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.353 + 0.935i$
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: \((2,\ 847,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516618 - 0.356895i\)
\(L(\frac12)\) \(\approx\) \(0.516618 - 0.356895i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.613 + 1.88i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.25 + 1.63i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.00832 - 0.0256i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.50 - 4.64i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.518 - 1.59i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.11 - 0.807i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 8.06T + 23T^{2} \)
29 \( 1 + (5.17 - 3.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.23 - 3.81i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.421 - 0.306i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.57 + 6.22i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.73T + 43T^{2} \)
47 \( 1 + (-7.24 - 5.26i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.22 + 3.76i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.87 - 5.71i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.61 - 8.05i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.81T + 67T^{2} \)
71 \( 1 + (-0.632 + 1.94i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.46 - 6.15i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.80 + 5.56i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.805 - 2.47i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.21T + 89T^{2} \)
97 \( 1 + (-1.04 - 3.23i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.65024937857437670060196351146, −9.277056307434424206018264481252, −8.656685237918491129128767534779, −7.18717071272829365003939660854, −6.74558910216687442514866130818, −5.63982475732208827608194414318, −4.60058189082930489928908992693, −3.30144358596570392599394744533, −1.77930878213350429628622189889, −1.12865955488084428567882182149, 0.51172428521840287852491741829, 3.12030565867261715236483968685, 4.68024827536725634749158437755, 5.25771508713784901910749567028, 5.95008438171523337447486824078, 6.74737980470050239128583290056, 7.70977759756589027348364965967, 8.623708374782776388337442128470, 9.430596557427752349085088803185, 10.26828135390986349782116410598

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