↑

Properties

Label 2-847-11.4-c1-0-45
Degree $2$
Conductor $847$
Sign $0.530 + 0.847i$
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  + (1.05 + 0.765i)2-s + (0.402 − 1.23i)3-s + (−0.0935 − 0.287i)4-s + (2.91 − 2.11i)5-s + (1.37 − 0.997i)6-s + (−0.309 − 0.951i)7-s + (0.927 − 2.85i)8-s + (1.05 + 0.765i)9-s + 4.69·10-s − 0.394·12-s + (0.489 + 0.355i)13-s + (0.402 − 1.23i)14-s + (−1.45 − 4.46i)15-s + (2.67 − 1.94i)16-s + (−5.09 + 3.70i)17-s + (0.524 + 1.61i)18-s + ⋯
L(s)  = 1  + (0.745 + 0.541i)2-s + (0.232 − 0.715i)3-s + (−0.0467 − 0.143i)4-s + (1.30 − 0.947i)5-s + (0.560 − 0.407i)6-s + (−0.116 − 0.359i)7-s + (0.327 − 1.00i)8-s + (0.351 + 0.255i)9-s + 1.48·10-s − 0.113·12-s + (0.135 + 0.0987i)13-s + (0.107 − 0.331i)14-s + (−0.374 − 1.15i)15-s + (0.668 − 0.485i)16-s + (−1.23 + 0.898i)17-s + (0.123 + 0.380i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59785 - 1.43954i\)
\(L(\frac12)\) \(\approx\) \(2.59785 - 1.43954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-1.05 - 0.765i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (-0.402 + 1.23i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.91 + 2.11i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (-0.489 - 0.355i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.09 - 3.70i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.927 - 2.85i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 + (-2.56 - 7.89i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.84 - 8.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.16 + 6.65i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 5.30T + 43T^{2} \)
47 \( 1 + (-2.75 + 8.47i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.69 - 1.22i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.18 - 9.79i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.564 + 0.409i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 9.51T + 67T^{2} \)
71 \( 1 + (-0.564 + 0.409i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.54 + 4.75i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.80 - 2.76i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.42 - 1.76i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + (2.91 + 2.11i)T + (29.9 + 92.2i)T^{2} \)
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

−10.17366755622922718284620158335, −9.107430773911077786199210549343, −8.354129517613534717231065761560, −7.20592754023577904482638443620, −6.37081348067203953770306433275, −5.81075431096755283702015828390, −4.79668990787731762481585531604, −4.02660361688144930394977815388, −2.08700668011813504667069307735, −1.26286834348546849073793176061, 2.24376205760886122782545030711, 2.73849634178413276158584218202, 3.94572537754655585310397340687, 4.72568035608986031035195298637, 5.86147621493083170609153622592, 6.59656002677328737469799363825, 7.76664098511971100720079316243, 9.069300281151153279557692563476, 9.519316086502628660809402487698, 10.42736276035165079715692398426

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