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Properties

Label 2-847-11.5-c1-0-39
Degree $2$
Conductor $847$
Sign $0.353 + 0.935i$
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  + (0.527 + 1.62i)2-s + (−1.85 − 1.34i)3-s + (−0.739 + 0.537i)4-s + (1.25 − 3.86i)5-s + (1.20 − 3.72i)6-s + (0.809 − 0.587i)7-s + (1.49 + 1.08i)8-s + (0.695 + 2.14i)9-s + 6.94·10-s + 2.09·12-s + (−1.01 − 3.11i)13-s + (1.38 + 1.00i)14-s + (−7.53 + 5.47i)15-s + (−1.54 + 4.74i)16-s + (0.408 − 1.25i)17-s + (−3.10 + 2.25i)18-s + ⋯
L(s)  = 1  + (0.373 + 1.14i)2-s + (−1.07 − 0.777i)3-s + (−0.369 + 0.268i)4-s + (0.561 − 1.72i)5-s + (0.493 − 1.51i)6-s + (0.305 − 0.222i)7-s + (0.530 + 0.385i)8-s + (0.231 + 0.713i)9-s + 2.19·10-s + 0.604·12-s + (−0.280 − 0.864i)13-s + (0.369 + 0.268i)14-s + (−1.94 + 1.41i)15-s + (−0.385 + 1.18i)16-s + (0.0989 − 0.304i)17-s + (−0.732 + 0.532i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18191 - 0.816499i\)
\(L(\frac12)\) \(\approx\) \(1.18191 - 0.816499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.527 - 1.62i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.85 + 1.34i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.25 + 3.86i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.01 + 3.11i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.408 + 1.25i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.74 - 1.27i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.86T + 23T^{2} \)
29 \( 1 + (0.197 - 0.143i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.12 + 6.53i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.206 - 0.150i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (4.63 + 3.36i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.01T + 43T^{2} \)
47 \( 1 + (-3.29 - 2.39i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (1.54 + 4.76i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.795 + 0.578i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.570 - 1.75i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 3.00T + 67T^{2} \)
71 \( 1 + (-2.00 + 6.15i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.80 - 5.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-1.71 - 5.26i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.639 - 1.96i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.6T + 89T^{2} \)
97 \( 1 + (0.798 + 2.45i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.958789156470011318108596382007, −8.931382261511002144405148066134, −7.956681033516795849141637687126, −7.42764323861825323512529605256, −6.29888996751502774239427185903, −5.55646570401737454518868548926, −5.26155691195585029543679907725, −4.27565708473373801874432775619, −1.84313639829240507514860273639, −0.71485306694356436922930779841, 1.87351207836493914157406071735, 2.87380268875995523022353106913, 3.86260667515669418409279589040, 4.84474265029153667821280569926, 5.86117431099163220784904530790, 6.71744241136041510458142763963, 7.52185220058820759537011291594, 9.245766372072063134757712232010, 10.08329945026829903252627586633, 10.54662272024363788442369031584

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