Properties

Label 2-847-77.76-c1-0-53
Degree $2$
Conductor $847$
Sign $-0.563 - 0.826i$
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  − 2.06i·2-s − 1.78i·3-s − 2.25·4-s − 0.290i·5-s − 3.68·6-s + (2.64 − 0.129i)7-s + 0.517i·8-s − 0.200·9-s − 0.598·10-s + 4.02i·12-s − 2.85·13-s + (−0.267 − 5.44i)14-s − 0.519·15-s − 3.43·16-s − 6.77·17-s + 0.413i·18-s + ⋯
L(s)  = 1  − 1.45i·2-s − 1.03i·3-s − 1.12·4-s − 0.129i·5-s − 1.50·6-s + (0.998 − 0.0491i)7-s + 0.183i·8-s − 0.0668·9-s − 0.189·10-s + 1.16i·12-s − 0.792·13-s + (−0.0716 − 1.45i)14-s − 0.134·15-s − 0.858·16-s − 1.64·17-s + 0.0975i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.563 - 0.826i$
Analytic conductor: \(6.76332\)
Root analytic conductor: \(2.60064\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (846, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :1/2),\ -0.563 - 0.826i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625667 + 1.18411i\)
\(L(\frac12)\) \(\approx\) \(0.625667 + 1.18411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.64 + 0.129i)T \)
11 \( 1 \)
good2 \( 1 + 2.06iT - 2T^{2} \)
3 \( 1 + 1.78iT - 3T^{2} \)
5 \( 1 + 0.290iT - 5T^{2} \)
13 \( 1 + 2.85T + 13T^{2} \)
17 \( 1 + 6.77T + 17T^{2} \)
19 \( 1 + 2.49T + 19T^{2} \)
23 \( 1 + 4.08T + 23T^{2} \)
29 \( 1 + 6.91iT - 29T^{2} \)
31 \( 1 + 5.12iT - 31T^{2} \)
37 \( 1 - 6.31T + 37T^{2} \)
41 \( 1 + 9.51T + 41T^{2} \)
43 \( 1 + 2.51iT - 43T^{2} \)
47 \( 1 - 7.65iT - 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 4.12iT - 59T^{2} \)
61 \( 1 - 6.06T + 61T^{2} \)
67 \( 1 - 2.66T + 67T^{2} \)
71 \( 1 - 5.53T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 13.9iT - 79T^{2} \)
83 \( 1 - 0.375T + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + 5.85iT - 97T^{2} \)
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

−9.887678033390361239429021961087, −8.883658574562481942456235309205, −8.051204725110290569894793720210, −7.14531147972975068567339583919, −6.25988522589761779569366444189, −4.72200610195602199316455156200, −4.12236231751854803261730634241, −2.37070622139410203045444688495, −2.00964454852610727201123828375, −0.63468769728626880953916169641, 2.22472132535430933846829870263, 3.98733970598967443820066096368, 4.87407368499258826072446762430, 5.22153186130394502818980321189, 6.64428316202179545603435546338, 7.11764149004883324173238343461, 8.334880087297304729130805352357, 8.726857299419914857976360571744, 9.708423918634411456975335163574, 10.69385238487210636865159741466

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