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Properties

Label 2-847-11.4-c1-0-0
Degree $2$
Conductor $847$
Sign $-0.975 + 0.220i$
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  + (−2.18 − 1.58i)2-s + (−0.867 + 2.66i)3-s + (1.64 + 5.04i)4-s + (0.360 − 0.261i)5-s + (6.13 − 4.46i)6-s + (0.309 + 0.951i)7-s + (2.76 − 8.50i)8-s + (−3.94 − 2.86i)9-s − 1.20·10-s − 14.8·12-s + (−0.364 − 0.265i)13-s + (0.835 − 2.57i)14-s + (0.386 + 1.18i)15-s + (−10.9 + 7.96i)16-s + (−3.90 + 2.83i)17-s + (4.07 + 12.5i)18-s + ⋯
L(s)  = 1  + (−1.54 − 1.12i)2-s + (−0.500 + 1.54i)3-s + (0.820 + 2.52i)4-s + (0.161 − 0.116i)5-s + (2.50 − 1.82i)6-s + (0.116 + 0.359i)7-s + (0.976 − 3.00i)8-s + (−1.31 − 0.956i)9-s − 0.380·10-s − 4.30·12-s + (−0.101 − 0.0735i)13-s + (0.223 − 0.687i)14-s + (0.0996 + 0.306i)15-s + (−2.74 + 1.99i)16-s + (−0.947 + 0.688i)17-s + (0.960 + 2.95i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.975 + 0.220i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0119274 - 0.106765i\)
\(L(\frac12)\) \(\approx\) \(0.0119274 - 0.106765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 1.58i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.867 - 2.66i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.360 + 0.261i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (0.364 + 0.265i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.90 - 2.83i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.335 - 1.03i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.57T + 23T^{2} \)
29 \( 1 + (0.613 + 1.88i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.68 + 4.85i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.26 - 6.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.547 + 1.68i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + (-0.315 + 0.972i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.88 - 2.09i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.44 + 13.6i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.98 - 2.89i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 + (-4.79 + 3.48i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.512 + 1.57i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.91 - 2.12i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.74 + 6.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.21T + 89T^{2} \)
97 \( 1 + (-4.29 - 3.12i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.53053405472021168249724587979, −9.832246588601086656458665220670, −9.219142982429777080705374515399, −8.679318726601041797592490525572, −7.68701345278742351665040647582, −6.40581333938970891297195613819, −5.10408801146832155102252453588, −4.01317440718989665728604339604, −3.16143583100785863126398729849, −1.82399271158008973093450752573, 0.098087580831008067910494081516, 1.28538910638404859907114673898, 2.33753180529087762257487355049, 4.97302947168628427261292999987, 5.89057041973755322014256755823, 6.79264069964254595546140329042, 7.05940431457099959134521029403, 7.83926821075408554467919124377, 8.695636377293794960527189887480, 9.399660686567797316123575428726

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