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Properties

Label 2-847-11.9-c1-0-18
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  + (−0.0699 + 0.215i)2-s + (0.177 − 0.128i)3-s + (1.57 + 1.14i)4-s + (−0.771 − 2.37i)5-s + (0.0153 + 0.0471i)6-s + (0.809 + 0.587i)7-s + (−0.722 + 0.525i)8-s + (−0.912 + 2.80i)9-s + 0.564·10-s + 0.427·12-s + (−1.58 + 4.88i)13-s + (−0.183 + 0.132i)14-s + (−0.442 − 0.321i)15-s + (1.14 + 3.51i)16-s + (0.444 + 1.36i)17-s + (−0.540 − 0.392i)18-s + ⋯
L(s)  = 1  + (−0.0494 + 0.152i)2-s + (0.102 − 0.0743i)3-s + (0.788 + 0.572i)4-s + (−0.344 − 1.06i)5-s + (0.00625 + 0.0192i)6-s + (0.305 + 0.222i)7-s + (−0.255 + 0.185i)8-s + (−0.304 + 0.935i)9-s + 0.178·10-s + 0.123·12-s + (−0.440 + 1.35i)13-s + (−0.0489 + 0.0355i)14-s + (−0.114 − 0.0829i)15-s + (0.285 + 0.878i)16-s + (0.107 + 0.331i)17-s + (−0.127 − 0.0925i)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: $\chi_{847} (372, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.353 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35988 + 0.939449i\)
\(L(\frac12)\) \(\approx\) \(1.35988 + 0.939449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0699 - 0.215i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 + (-0.177 + 0.128i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.771 + 2.37i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (1.58 - 4.88i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.444 - 1.36i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.90 - 3.56i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.08T + 23T^{2} \)
29 \( 1 + (-5.27 - 3.83i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.37 + 7.31i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-3.22 - 2.34i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.46 + 3.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 0.802T + 43T^{2} \)
47 \( 1 + (5.46 - 3.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.03 + 6.25i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (2.32 + 1.69i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.264 - 0.813i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 1.64T + 67T^{2} \)
71 \( 1 + (-1.39 - 4.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (12.0 + 8.72i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (0.757 - 2.33i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.694 + 2.13i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 1.73T + 89T^{2} \)
97 \( 1 + (3.73 - 11.4i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.52938711653543098558025064510, −9.214278692078424411775660428606, −8.469213426860383850450479654374, −7.982856498479426980047689673427, −7.03485692775173244857099294677, −6.08453782983986293886988049360, −4.89819948461123694555052688158, −4.17929800352268597291834937499, −2.67009731442544118320455676745, −1.66210071837211361695198009305, 0.818860908107798424483135732163, 2.73783737340360321626328338345, 3.08981504013415219060812414142, 4.64986560073386998216489394896, 5.81465781922751319875003716564, 6.72372744220298891124533408681, 7.20361615696382673291182221313, 8.285988392320780929266549506462, 9.345036158046594262377620823185, 10.32147997902576991257816427384

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