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Properties

Label 2-847-11.5-c1-0-7
Degree $2$
Conductor $847$
Sign $-0.780 + 0.625i$
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  + (−0.690 − 2.12i)2-s + (−2.61 − 1.90i)3-s + (−2.42 + 1.76i)4-s + (−0.618 + 1.90i)5-s + (−2.23 + 6.88i)6-s + (−0.809 + 0.587i)7-s + (1.80 + 1.31i)8-s + (2.30 + 7.10i)9-s + 4.47·10-s + 9.70·12-s + (−0.381 − 1.17i)13-s + (1.80 + 1.31i)14-s + (5.23 − 3.80i)15-s + (−0.309 + 0.951i)16-s + (0.381 − 1.17i)17-s + (13.5 − 9.82i)18-s + ⋯
L(s)  = 1  + (−0.488 − 1.50i)2-s + (−1.51 − 1.09i)3-s + (−1.21 + 0.881i)4-s + (−0.276 + 0.850i)5-s + (−0.912 + 2.80i)6-s + (−0.305 + 0.222i)7-s + (0.639 + 0.464i)8-s + (0.769 + 2.36i)9-s + 1.41·10-s + 2.80·12-s + (−0.105 − 0.326i)13-s + (0.483 + 0.351i)14-s + (1.35 − 0.982i)15-s + (−0.0772 + 0.237i)16-s + (0.0926 − 0.285i)17-s + (3.18 − 2.31i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.780 + 0.625i$
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: \((2,\ 847,\ (\ :1/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.145772 - 0.414818i\)
\(L(\frac12)\) \(\approx\) \(0.145772 - 0.414818i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.690 + 2.12i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (2.61 + 1.90i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.618 - 1.90i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.381 + 1.17i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.381 + 1.17i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2 - 1.45i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6.47T + 23T^{2} \)
29 \( 1 + (-0.381 + 0.277i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.23 + 6.88i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.381 - 0.277i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-5.47 - 3.97i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + (5.85 + 4.25i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.61 - 8.05i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.61 - 1.90i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.854 - 2.62i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + (0.472 - 1.45i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.23 + 3.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.76 - 8.50i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.76 + 14.6i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + (2.90 + 8.95i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.27122488694674165311454741011, −9.461940989180405661935382081546, −8.033298902128904743599989032174, −7.34483793381457420397964355608, −6.33317901372038733043231135328, −5.63201656526220505204101478447, −4.21329748203579252280791607550, −2.89200360225507828187062921631, −1.88560124352475283127409024985, −0.56297473992152326454418960087, 0.66789905647688990161723217148, 3.83042734037164720052972105621, 4.74110483392989472856492209123, 5.34288884267901162165556214471, 6.16625190077651315220875092964, 6.84040434682005930474360523051, 7.903753999959053492153102265401, 8.941868049470362119762868305618, 9.512461293108285538852625637985, 10.28213937943088419472172092534

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