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Properties

Label 2-847-11.9-c1-0-37
Degree $2$
Conductor $847$
Sign $-0.923 + 0.382i$
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  + (0.651 − 2.00i)2-s + (−1.37 + 0.998i)3-s + (−1.98 − 1.43i)4-s + (0.152 + 0.468i)5-s + (1.10 + 3.40i)6-s + (−0.809 − 0.587i)7-s + (−0.767 + 0.557i)8-s + (−0.0355 + 0.109i)9-s + 1.03·10-s + 4.16·12-s + (1.63 − 5.04i)13-s + (−1.70 + 1.23i)14-s + (−0.676 − 0.491i)15-s + (−0.895 − 2.75i)16-s + (0.938 + 2.88i)17-s + (0.196 + 0.142i)18-s + ⋯
L(s)  = 1  + (0.460 − 1.41i)2-s + (−0.793 + 0.576i)3-s + (−0.990 − 0.719i)4-s + (0.0680 + 0.209i)5-s + (0.451 + 1.39i)6-s + (−0.305 − 0.222i)7-s + (−0.271 + 0.197i)8-s + (−0.0118 + 0.0364i)9-s + 0.328·10-s + 1.20·12-s + (0.454 − 1.39i)13-s + (−0.456 + 0.331i)14-s + (−0.174 − 0.126i)15-s + (−0.223 − 0.689i)16-s + (0.227 + 0.700i)17-s + (0.0462 + 0.0336i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.923 + 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.228584 - 1.14940i\)
\(L(\frac12)\) \(\approx\) \(0.228584 - 1.14940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.651 + 2.00i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.37 - 0.998i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.152 - 0.468i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (-1.63 + 5.04i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.938 - 2.88i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.77 + 2.74i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.63T + 23T^{2} \)
29 \( 1 + (5.60 + 4.06i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.391 - 1.20i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.79 + 6.39i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.16 + 0.848i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 2.88T + 43T^{2} \)
47 \( 1 + (-7.08 + 5.14i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.05 + 6.31i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.76 - 4.91i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.29 + 13.2i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 + (-1.83 - 5.65i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.05 - 2.21i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (2.72 - 8.37i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-3.42 - 10.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 3.10T + 89T^{2} \)
97 \( 1 + (1.95 - 6.00i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.29896595259267870031045556284, −9.546749994707103750444137945555, −8.279702108945651494665557825884, −7.19660325516120562029619037942, −5.82317388506944922075539461280, −5.27137743394063100007067629793, −4.11350010108733889133330365432, −3.39199236542847677818709646321, −2.22372165067005559662728696970, −0.56159445068379476745357539887, 1.55580166595665942822472417787, 3.55916784484170161857903712122, 4.71785966979988824147545319666, 5.67237500299107965993613711615, 6.13011549600611083142000061243, 7.04634556742402824057938344917, 7.49335303618628880822968146193, 8.774672321095349178645688041847, 9.332015146963059440823058278066, 10.65211211624320808594573532299

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