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Properties

Label 2-847-77.76-c1-0-38
Degree $2$
Conductor $847$
Sign $0.991 - 0.127i$
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  − 0.129i·2-s + 2.08i·3-s + 1.98·4-s − 1.78i·5-s + 0.270·6-s + (−1.65 − 2.06i)7-s − 0.517i·8-s − 1.33·9-s − 0.231·10-s + 4.12i·12-s − 3.79·13-s + (−0.267 + 0.215i)14-s + 3.71·15-s + 3.89·16-s + 5.68·17-s + 0.173i·18-s + ⋯
L(s)  = 1  − 0.0918i·2-s + 1.20i·3-s + 0.991·4-s − 0.797i·5-s + 0.110·6-s + (−0.626 − 0.779i)7-s − 0.183i·8-s − 0.445·9-s − 0.0733·10-s + 1.19i·12-s − 1.05·13-s + (−0.0716 + 0.0575i)14-s + 0.959·15-s + 0.974·16-s + 1.37·17-s + 0.0408i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.127i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.991 - 0.127i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.991 - 0.127i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94432 + 0.124407i\)
\(L(\frac12)\) \(\approx\) \(1.94432 + 0.124407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.65 + 2.06i)T \)
11 \( 1 \)
good2 \( 1 + 0.129iT - 2T^{2} \)
3 \( 1 - 2.08iT - 3T^{2} \)
5 \( 1 + 1.78iT - 5T^{2} \)
13 \( 1 + 3.79T + 13T^{2} \)
17 \( 1 - 5.68T + 17T^{2} \)
19 \( 1 - 4.66T + 19T^{2} \)
23 \( 1 - 6.35T + 23T^{2} \)
29 \( 1 - 3.12iT - 29T^{2} \)
31 \( 1 + 1.29iT - 31T^{2} \)
37 \( 1 - 0.949T + 37T^{2} \)
41 \( 1 + 2.70T + 41T^{2} \)
43 \( 1 + 7.27iT - 43T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 - 8.25iT - 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 3.79T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 + 4.61T + 73T^{2} \)
79 \( 1 + 0.183iT - 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 3.59iT - 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−10.04964735362872090202637263397, −9.747004398748898715268456479164, −8.732981468756219222043453727608, −7.43976732868938354021852908519, −6.98453867842573714648738608069, −5.50495592697650107506908752768, −4.93061763631595944155853275866, −3.70110436147470898925482562268, −2.98124701591332685087059625072, −1.13469529173669702194583751725, 1.35645404719223671626181960535, 2.72237465461887153336299530582, 3.05874209369698180291015396256, 5.21432808420448439550836264520, 6.14630175499258305383721877373, 6.81702397052490797565314716240, 7.42802330210679830386816861225, 8.036054675791803829022103384565, 9.495171541525558800594924856434, 10.12586861059733285474993595808

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