Properties

Label 2-847-11.4-c1-0-32
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  + (−0.666 − 0.484i)2-s + (−0.296 + 0.913i)3-s + (−0.408 − 1.25i)4-s + (−2.41 + 1.75i)5-s + (0.640 − 0.465i)6-s + (−0.309 − 0.951i)7-s + (−0.845 + 2.60i)8-s + (1.68 + 1.22i)9-s + 2.45·10-s + 1.26·12-s + (1.78 + 1.29i)13-s + (−0.254 + 0.783i)14-s + (−0.886 − 2.72i)15-s + (−0.315 + 0.229i)16-s + (3.56 − 2.59i)17-s + (−0.528 − 1.62i)18-s + ⋯
L(s)  = 1  + (−0.471 − 0.342i)2-s + (−0.171 + 0.527i)3-s + (−0.204 − 0.628i)4-s + (−1.08 + 0.784i)5-s + (0.261 − 0.189i)6-s + (−0.116 − 0.359i)7-s + (−0.298 + 0.919i)8-s + (0.560 + 0.406i)9-s + 0.777·10-s + 0.366·12-s + (0.493 + 0.358i)13-s + (−0.0680 + 0.209i)14-s + (−0.228 − 0.704i)15-s + (−0.0789 + 0.0573i)16-s + (0.864 − 0.628i)17-s + (−0.124 − 0.383i)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

\(L(1)\) \(\approx\) \(0.0362327 - 0.157094i\)
\(L(\frac12)\) \(\approx\) \(0.0362327 - 0.157094i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.666 + 0.484i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.296 - 0.913i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (2.41 - 1.75i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-1.78 - 1.29i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.56 + 2.59i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.533 + 1.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 8.39T + 23T^{2} \)
29 \( 1 + (1.01 + 3.13i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.04 + 4.39i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.71 + 8.35i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.66 - 5.12i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.44T + 43T^{2} \)
47 \( 1 + (1.66 - 5.13i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (7.60 + 5.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.07 - 3.30i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-10.5 + 7.63i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.32T + 67T^{2} \)
71 \( 1 + (3.56 - 2.58i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.55 + 14.0i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.81 + 4.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.17 + 4.48i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (2.30 + 1.67i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.954334543214873566970561981757, −9.317745655368642744054056907483, −8.052063632825600617443912346000, −7.48425626507179439994729736637, −6.38282289493389631915521784397, −5.29861866769133794330761126499, −4.25226883681189601109513191975, −3.46073417090502561351592897910, −1.89107253354865290435878085356, −0.10137829690085258235554185073, 1.39402643690917262808093111217, 3.50276530104497930148179988745, 3.98772864778399415561186909274, 5.35123513843522143036813390296, 6.48914937232379297820454535770, 7.32847036945320314613250408382, 8.247830113775132372723445209259, 8.378926948085799403814689447619, 9.541589748368930793030104606812, 10.35341010178266757597247538874

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