↑

Properties

Label 2-847-847.356-c0-0-0
Degree $2$
Conductor $847$
Sign $0.914 + 0.403i$
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  + (−0.391 + 1.93i)2-s + (−2.65 − 1.12i)4-s + (−0.985 + 0.170i)7-s + (2.09 − 3.06i)8-s + (−0.809 − 0.587i)9-s + (−0.870 − 0.491i)11-s + (0.0562 − 1.96i)14-s + (3.08 + 3.17i)16-s + (1.45 − 1.33i)18-s + (1.29 − 1.48i)22-s + (−0.255 − 1.77i)23-s + (−0.254 − 0.967i)25-s + (2.80 + 0.653i)28-s + (−0.310 + 1.18i)29-s + (−4.22 + 2.71i)32-s + ⋯
L(s)  = 1  + (−0.391 + 1.93i)2-s + (−2.65 − 1.12i)4-s + (−0.985 + 0.170i)7-s + (2.09 − 3.06i)8-s + (−0.809 − 0.587i)9-s + (−0.870 − 0.491i)11-s + (0.0562 − 1.96i)14-s + (3.08 + 3.17i)16-s + (1.45 − 1.33i)18-s + (1.29 − 1.48i)22-s + (−0.255 − 1.77i)23-s + (−0.254 − 0.967i)25-s + (2.80 + 0.653i)28-s + (−0.310 + 1.18i)29-s + (−4.22 + 2.71i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (356, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.914 + 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2055587970\)
\(L(\frac12)\) \(\approx\) \(0.2055587970\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.985 - 0.170i)T \)
11 \( 1 + (0.870 + 0.491i)T \)
good2 \( 1 + (0.391 - 1.93i)T + (-0.921 - 0.389i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.254 + 0.967i)T^{2} \)
13 \( 1 + (-0.897 + 0.441i)T^{2} \)
17 \( 1 + (0.736 + 0.676i)T^{2} \)
19 \( 1 + (-0.198 + 0.980i)T^{2} \)
23 \( 1 + (0.255 + 1.77i)T + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (0.310 - 1.18i)T + (-0.870 - 0.491i)T^{2} \)
31 \( 1 + (0.985 - 0.170i)T^{2} \)
37 \( 1 + (-0.319 + 0.529i)T + (-0.466 - 0.884i)T^{2} \)
41 \( 1 + (-0.974 + 0.226i)T^{2} \)
43 \( 1 + (1.33 + 0.392i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + (-0.0855 - 0.996i)T^{2} \)
53 \( 1 + (1.39 - 1.43i)T + (-0.0285 - 0.999i)T^{2} \)
59 \( 1 + (-0.974 - 0.226i)T^{2} \)
61 \( 1 + (0.921 - 0.389i)T^{2} \)
67 \( 1 + (0.301 + 0.660i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (1.94 - 0.111i)T + (0.993 - 0.113i)T^{2} \)
73 \( 1 + (-0.610 + 0.791i)T^{2} \)
79 \( 1 + (-0.161 - 0.0575i)T + (0.774 + 0.633i)T^{2} \)
83 \( 1 + (0.564 - 0.825i)T^{2} \)
89 \( 1 + (-0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.254 - 0.967i)T^{2} \)
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

−10.03057670041244517938912480202, −9.052788470541685740641868398749, −8.600458440160262988167289331985, −7.78627346304315510128006573818, −6.73092405125789588379572829321, −6.17546429781975722209822451176, −5.49621048086998239244495389327, −4.41215698791392899666710617228, −3.11907671099406495166679990170, −0.23034595860646051389239846698, 1.82517675155935646380998969322, 2.90002417930045385538560916566, 3.59718321362664816432062928820, 4.81740109333891518130927429105, 5.71186429435863300157981159124, 7.51652736293260142299628727991, 8.161882431838936910405724891266, 9.223237836695490663303566579017, 9.825380294456941309940112997103, 10.36873307510174307619838606634

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