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Properties

Label 2-847-11.9-c1-0-52
Degree $2$
Conductor $847$
Sign $-0.569 - 0.821i$
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  + (0.762 − 2.34i)2-s + (1.30 − 0.951i)3-s + (−3.30 − 2.40i)4-s + (−1.07 − 3.29i)5-s + (−1.23 − 3.79i)6-s + (0.809 + 0.587i)7-s + (−4.16 + 3.02i)8-s + (−0.118 + 0.363i)9-s − 8.55·10-s − 6.60·12-s + (−0.202 + 0.621i)13-s + (1.99 − 1.44i)14-s + (−4.53 − 3.29i)15-s + (1.39 + 4.29i)16-s + (−0.351 − 1.08i)17-s + (0.762 + 0.553i)18-s + ⋯
L(s)  = 1  + (0.539 − 1.65i)2-s + (0.755 − 0.549i)3-s + (−1.65 − 1.20i)4-s + (−0.479 − 1.47i)5-s + (−0.503 − 1.54i)6-s + (0.305 + 0.222i)7-s + (−1.47 + 1.06i)8-s + (−0.0393 + 0.121i)9-s − 2.70·10-s − 1.90·12-s + (−0.0560 + 0.172i)13-s + (0.533 − 0.387i)14-s + (−1.17 − 0.851i)15-s + (0.348 + 1.07i)16-s + (−0.0852 − 0.262i)17-s + (0.179 + 0.130i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.569 - 0.821i$
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: \((2,\ 847,\ (\ :1/2),\ -0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.962319 + 1.83841i\)
\(L(\frac12)\) \(\approx\) \(0.962319 + 1.83841i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.762 + 2.34i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-1.30 + 0.951i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.07 + 3.29i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.202 - 0.621i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.351 + 1.08i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.91 + 3.57i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 6.66T + 23T^{2} \)
29 \( 1 + (-3.70 - 2.68i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.864 + 2.65i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.355 - 0.258i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.77 + 3.46i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + (-0.489 + 0.355i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.03 + 9.34i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (1.37 + 0.995i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.11 - 6.52i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 6.17T + 67T^{2} \)
71 \( 1 + (1.67 + 5.15i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.42 + 3.93i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-0.820 + 2.52i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.06 - 6.36i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.698T + 89T^{2} \)
97 \( 1 + (4.59 - 14.1i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.558194330431252388999252244542, −8.995670044231612637399081314895, −8.253210587983993932059580458412, −7.40500971067795668695629691742, −5.56492288272739341863036709888, −4.80121787156770846800488029862, −4.03485183974874501297179507899, −2.83703192614371932549059205446, −1.88804626981884752605416382560, −0.814504688715402890006025281401, 2.83115382758914633017319406974, 3.75456235363800616077346849307, 4.36625927255144862149151990581, 5.79336311894160442632426472231, 6.40477907331368464004129934868, 7.47323902593082690098736472027, 7.81816550970170657087287014740, 8.690527729543243352428868959161, 9.760304902477013643262557482963, 10.48139973942920569574806102776

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