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Properties

Label 2-847-11.9-c1-0-46
Degree $2$
Conductor $847$
Sign $0.437 + 0.899i$
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  + (2.42 − 1.76i)3-s + (1.61 + 1.17i)4-s + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)7-s + (1.85 − 5.70i)9-s + 6·12-s + (1.23 − 3.80i)13-s + (−2.42 − 1.76i)15-s + (1.23 + 3.80i)16-s + (−0.618 − 1.90i)17-s + (−4.85 + 3.52i)19-s + (0.618 − 1.90i)20-s − 3·21-s − 5·23-s + (3.23 − 2.35i)25-s + ⋯
L(s)  = 1  + (1.40 − 1.01i)3-s + (0.809 + 0.587i)4-s + (−0.138 − 0.425i)5-s + (−0.305 − 0.222i)7-s + (0.618 − 1.90i)9-s + 1.73·12-s + (0.342 − 1.05i)13-s + (−0.626 − 0.455i)15-s + (0.309 + 0.951i)16-s + (−0.149 − 0.461i)17-s + (−1.11 + 0.809i)19-s + (0.138 − 0.425i)20-s − 0.654·21-s − 1.04·23-s + (0.647 − 0.470i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.437 + 0.899i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.437 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37170 - 1.48292i\)
\(L(\frac12)\) \(\approx\) \(2.37170 - 1.48292i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-2.42 + 1.76i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.309 + 0.951i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.23 + 3.80i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.618 + 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.85 - 3.52i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5T + 23T^{2} \)
29 \( 1 + (-8.09 - 5.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.04 - 2.93i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.61 - 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (6.47 - 4.70i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.85 - 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.42 + 1.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.618 - 1.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 3T + 67T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.09 - 5.87i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.85 - 5.70i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (3.70 + 11.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
97 \( 1 + (1.54 - 4.75i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.970960891042051973000426335792, −8.752534845451673185531178972594, −8.226540829701649048991908317678, −7.71783801418545149448320001522, −6.76930020858413221706683836030, −6.10074884962540342783625618715, −4.27340117459145478232609343687, −3.20103526466802375520514475617, −2.53766212771330727118091720749, −1.27831561270769580785015426772, 2.05131082770292051708494333216, 2.75878029288106046887440356866, 3.87159897017269670202448671487, 4.71158784166738734291601555283, 6.16283522315935989065843124132, 6.86080151981069932241731810699, 7.995455202061340822625152718145, 8.777469420663855341100451389518, 9.540187014334091998859021516652, 10.24938310000116205102738264675

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