↑

Properties

Label 2-847-847.146-c0-0-0
Degree $2$
Conductor $847$
Sign $0.960 + 0.276i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.237 + 0.449i)2-s + (0.418 − 0.612i)4-s + (−0.254 − 0.967i)7-s + (0.879 + 0.100i)8-s + (−0.809 + 0.587i)9-s + (0.696 − 0.717i)11-s + (0.374 − 0.343i)14-s + (−0.105 − 0.271i)16-s + (−0.456 − 0.224i)18-s + (0.487 + 0.143i)22-s + (1.30 + 0.837i)23-s + (−0.921 − 0.389i)25-s + (−0.698 − 0.249i)28-s + (−0.365 + 0.154i)29-s + (0.676 − 0.781i)32-s + ⋯
L(s)  = 1  + (0.237 + 0.449i)2-s + (0.418 − 0.612i)4-s + (−0.254 − 0.967i)7-s + (0.879 + 0.100i)8-s + (−0.809 + 0.587i)9-s + (0.696 − 0.717i)11-s + (0.374 − 0.343i)14-s + (−0.105 − 0.271i)16-s + (−0.456 − 0.224i)18-s + (0.487 + 0.143i)22-s + (1.30 + 0.837i)23-s + (−0.921 − 0.389i)25-s + (−0.698 − 0.249i)28-s + (−0.365 + 0.154i)29-s + (0.676 − 0.781i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.960 + 0.276i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.960 + 0.276i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.211221608\)
\(L(\frac12)\) \(\approx\) \(1.211221608\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.254 + 0.967i)T \)
11 \( 1 + (-0.696 + 0.717i)T \)
good2 \( 1 + (-0.237 - 0.449i)T + (-0.564 + 0.825i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (0.921 + 0.389i)T^{2} \)
13 \( 1 + (-0.774 + 0.633i)T^{2} \)
17 \( 1 + (-0.897 + 0.441i)T^{2} \)
19 \( 1 + (0.466 + 0.884i)T^{2} \)
23 \( 1 + (-1.30 - 0.837i)T + (0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.365 - 0.154i)T + (0.696 - 0.717i)T^{2} \)
31 \( 1 + (0.254 + 0.967i)T^{2} \)
37 \( 1 + (0.0176 - 0.617i)T + (-0.998 - 0.0570i)T^{2} \)
41 \( 1 + (-0.941 + 0.336i)T^{2} \)
43 \( 1 + (0.301 - 0.660i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.610 + 0.791i)T^{2} \)
53 \( 1 + (0.0620 - 0.159i)T + (-0.736 - 0.676i)T^{2} \)
59 \( 1 + (-0.941 - 0.336i)T^{2} \)
61 \( 1 + (0.564 + 0.825i)T^{2} \)
67 \( 1 + (0.277 - 1.92i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.161 - 1.87i)T + (-0.985 + 0.170i)T^{2} \)
73 \( 1 + (-0.198 + 0.980i)T^{2} \)
79 \( 1 + (1.06 + 0.600i)T + (0.516 + 0.856i)T^{2} \)
83 \( 1 + (-0.993 - 0.113i)T^{2} \)
89 \( 1 + (0.142 + 0.989i)T^{2} \)
97 \( 1 + (0.921 - 0.389i)T^{2} \)
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

−10.40542859535293350066692631299, −9.628997223916189946292645990406, −8.567015128366250370273741926389, −7.60757544540767400244000088545, −6.86449504606371232293433244830, −6.00583241902547346484078561059, −5.24394661420477505055031148034, −4.13045001956445892474865989009, −2.92720664660656384172277876083, −1.33986273797549612071652265271, 1.96005903720119502054696697936, 2.95936892617951805964849788826, 3.83543178319660319934863735832, 5.03876924993460101019434862132, 6.17048084160018200865317233389, 6.90964705042172775809378233095, 7.933544977964176057927192242460, 8.916231923346006560593275972927, 9.427921005509863620631512212755, 10.64017825870225823417065085166

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