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Properties

Label 2-847-11.4-c1-0-27
Degree $2$
Conductor $847$
Sign $-0.0694 + 0.997i$
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  + (−1.43 − 1.04i)2-s + (0.190 − 0.587i)3-s + (0.358 + 1.10i)4-s + (2.24 − 1.63i)5-s + (−0.888 + 0.645i)6-s + (−0.309 − 0.951i)7-s + (−0.461 + 1.42i)8-s + (2.11 + 1.53i)9-s − 4.93·10-s + 0.716·12-s + (3.47 + 2.52i)13-s + (−0.549 + 1.69i)14-s + (−0.530 − 1.63i)15-s + (4.02 − 2.92i)16-s + (2.22 − 1.61i)17-s + (−1.43 − 4.42i)18-s + ⋯
L(s)  = 1  + (−1.01 − 0.738i)2-s + (0.110 − 0.339i)3-s + (0.179 + 0.551i)4-s + (1.00 − 0.730i)5-s + (−0.362 + 0.263i)6-s + (−0.116 − 0.359i)7-s + (−0.163 + 0.502i)8-s + (0.706 + 0.512i)9-s − 1.56·10-s + 0.206·12-s + (0.963 + 0.699i)13-s + (−0.146 + 0.451i)14-s + (−0.136 − 0.421i)15-s + (1.00 − 0.730i)16-s + (0.540 − 0.392i)17-s + (−0.338 − 1.04i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0694 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.0694 + 0.997i$
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: \((2,\ 847,\ (\ :1/2),\ -0.0694 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855059 - 0.916682i\)
\(L(\frac12)\) \(\approx\) \(0.855059 - 0.916682i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.43 + 1.04i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.190 + 0.587i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.24 + 1.63i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-3.47 - 2.52i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.22 + 1.61i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.599 + 1.84i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.37T + 23T^{2} \)
29 \( 1 + (-2.66 - 8.20i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-0.162 - 0.117i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.319 - 0.984i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.96 + 9.13i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + (4.03 - 12.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.15 - 2.29i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.64 + 8.13i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.799 + 0.581i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.41T + 67T^{2} \)
71 \( 1 + (-1.63 + 1.18i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.08 + 9.48i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.09 + 3.69i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.39 + 1.01i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (9.39 + 6.82i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.900109212531229262555635584926, −9.165874113529677046113976654975, −8.690298371494988760535875635010, −7.62101718912736980092612643466, −6.68808503221892980589226953141, −5.51486273037604094897039819973, −4.65063839501277266268459922689, −3.03883509324372908501173119260, −1.70870658201189320930174432050, −1.14399102676611758600818929582, 1.24450374876580014803841359465, 2.92026778860601825466831512335, 3.93664531565468987904684266231, 5.58437690202328850853706411542, 6.29580714542554616831526667069, 6.92789537087200238002132078358, 8.018257630708263962857515089850, 8.699160874469900709445635766510, 9.735468679772614586850230845009, 9.960898277380000078214640459359

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