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Properties

Label 2-847-11.4-c1-0-20
Degree $2$
Conductor $847$
Sign $-0.898 + 0.437i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.898 + 0.437i$
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: \((2,\ 847,\ (\ :1/2),\ -0.898 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.682875 - 2.96075i\)
\(L(\frac12)\) \(\approx\) \(0.682875 - 2.96075i\)
\(L(\frac{3}{2})\) 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

−11.00043604464865768195639698955, −9.748371128820255745909472844209, −9.028734807475057639129499230589, −7.79636098287373357281385128392, −6.98209462459942619932708560036, −5.89643986802691933413753779010, −5.29182861548918940371419512126, −4.63310137649993459975183450967, −3.76103663276024001532250785108, −3.03025684998065333260127879761, 1.00881720621415220568044254597, 2.02128763359742486345552103532, 2.93249398523820303333854955039, 4.13846154041868830875876429217, 5.57600629600479991911946940997, 5.79184594638026038154954256608, 6.74946171817525935778033252529, 7.61650244859916126796271392642, 9.020728830430328865597622600404, 10.25741578610023897648634104194

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