↑

Properties

Label 2-847-11.5-c1-0-23
Degree $2$
Conductor $847$
Sign $0.957 + 0.288i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.853 + 2.62i)2-s + (−1.42 − 1.03i)3-s + (−4.55 + 3.30i)4-s + (−0.811 + 2.49i)5-s + (1.50 − 4.62i)6-s + (−0.809 + 0.587i)7-s + (−8.10 − 5.88i)8-s + (0.0318 + 0.0980i)9-s − 7.25·10-s + 9.91·12-s + (−0.737 − 2.27i)13-s + (−2.23 − 1.62i)14-s + (3.74 − 2.71i)15-s + (5.06 − 15.5i)16-s + (−0.737 + 2.27i)17-s + (−0.230 + 0.167i)18-s + ⋯
L(s)  = 1  + (0.603 + 1.85i)2-s + (−0.822 − 0.597i)3-s + (−2.27 + 1.65i)4-s + (−0.362 + 1.11i)5-s + (0.613 − 1.88i)6-s + (−0.305 + 0.222i)7-s + (−2.86 − 2.08i)8-s + (0.0106 + 0.0326i)9-s − 2.29·10-s + 2.86·12-s + (−0.204 − 0.629i)13-s + (−0.597 − 0.433i)14-s + (0.966 − 0.702i)15-s + (1.26 − 3.89i)16-s + (−0.178 + 0.550i)17-s + (−0.0542 + 0.0394i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.957 + 0.288i$
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: \((2,\ 847,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0596443 - 0.00877577i\)
\(L(\frac12)\) \(\approx\) \(0.0596443 - 0.00877577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.853 - 2.62i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.42 + 1.03i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.811 - 2.49i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (0.737 + 2.27i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.737 - 2.27i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.39 - 1.01i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 0.626T + 23T^{2} \)
29 \( 1 + (1.39 - 1.01i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.691 + 2.12i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.57 + 4.05i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (8.40 + 6.10i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.25T + 43T^{2} \)
47 \( 1 + (5.16 + 3.75i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.85 + 8.79i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.42 + 1.03i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (3.20 - 9.87i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 6.42T + 67T^{2} \)
71 \( 1 + (-2.49 + 7.68i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (8.40 - 6.10i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.71 - 14.5i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.94 + 12.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (2.58 + 7.94i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.00435525189686687071773316054, −8.941775233527243244477730633205, −7.957949876679234461612873742740, −7.21532623629389448486738181050, −6.70516685562947736281058670374, −5.92410895826882665476413995126, −5.37598826070224872376484084426, −4.00392188683894063033867409041, −3.11105475224586666202496653018, −0.02972846269079666934581377742, 1.27636083639176425827050448044, 2.75913163431081999777609770536, 4.02172309535610620163555054019, 4.69696744041396142248841849042, 5.15352418622650805757353641000, 6.24471670220094069741487311326, 8.062033766877594425241618182646, 9.173810980210661150832998665759, 9.594556270915350751748413948997, 10.48394203599857788764438499526

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