↑

Properties

Label 2-847-77.13-c1-0-49
Degree $2$
Conductor $847$
Sign $0.339 - 0.940i$
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  + (−2.17 − 0.707i)2-s + (−1.46 − 2.02i)3-s + (2.61 + 1.90i)4-s + (−2.37 + 0.771i)5-s + (1.76 + 5.43i)6-s + (0.958 − 2.46i)7-s + (−1.66 − 2.28i)8-s + (−1.00 + 3.07i)9-s + 5.71·10-s − 8.08i·12-s + (1.34 − 4.15i)13-s + (−3.82 + 4.68i)14-s + (5.04 + 3.66i)15-s + (−2.44 − 7.51i)17-s + (4.35 − 5.99i)18-s + (−1.76 + 1.28i)19-s + ⋯
L(s)  = 1  + (−1.53 − 0.499i)2-s + (−0.847 − 1.16i)3-s + (1.30 + 0.951i)4-s + (−1.06 + 0.345i)5-s + (0.720 + 2.21i)6-s + (0.362 − 0.932i)7-s + (−0.587 − 0.809i)8-s + (−0.333 + 1.02i)9-s + 1.80·10-s − 2.33i·12-s + (0.374 − 1.15i)13-s + (−1.02 + 1.25i)14-s + (1.30 + 0.946i)15-s + (−0.591 − 1.82i)17-s + (1.02 − 1.41i)18-s + (−0.405 + 0.294i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.339 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.339 - 0.940i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ 0.339 - 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.104300 + 0.0732645i\)
\(L(\frac12)\) \(\approx\) \(0.104300 + 0.0732645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.958 + 2.46i)T \)
11 \( 1 \)
good2 \( 1 + (2.17 + 0.707i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.46 + 2.02i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (2.37 - 0.771i)T + (4.04 - 2.93i)T^{2} \)
13 \( 1 + (-1.34 + 4.15i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.44 + 7.51i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.76 - 1.28i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 + (-1.54 + 2.12i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.560 - 0.182i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.42 - 1.76i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.76 - 1.28i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 + (4.05 + 5.58i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.618 - 1.90i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-5.09 + 7.01i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.674 - 2.07i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 5.94T + 67T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.94 - 2.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.03 - 1.31i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.59 - 8.00i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 9.84iT - 89T^{2} \)
97 \( 1 + (-0.560 - 0.182i)T + (78.4 + 57.0i)T^{2} \)
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

−9.685287238827614506325702045793, −8.301864884989360742667241683243, −7.888616358166724566364632766716, −7.20506470764734186368955832525, −6.62260310924935545450468302239, −5.17544624070700867220279950425, −3.70079963830650584863462237168, −2.28192719089337468718464612465, −0.872761409195777568539058658481, −0.16190585209163445835857728098, 1.80423822709987483517674711821, 3.98507467782936617821315216609, 4.57045780652634181477581706977, 5.94546237827454897930200284242, 6.50076091125409253540523654370, 7.904672618017178599412128564404, 8.442942419739774882159053427803, 9.106280408516469063874334725320, 9.898549070847464591388752112966, 10.78315352367961043650216743066

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