Properties

Label 1-847-847.208-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.876 + 0.482i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.5 − 0.866i)3-s + (0.0475 + 0.998i)4-s + (0.327 − 0.945i)5-s + (−0.959 + 0.281i)6-s + (0.654 − 0.755i)8-s + (−0.5 − 0.866i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (−0.786 − 0.618i)17-s + (−0.235 + 0.971i)18-s + (−0.786 + 0.618i)19-s + (0.959 + 0.281i)20-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (0.5 − 0.866i)3-s + (0.0475 + 0.998i)4-s + (0.327 − 0.945i)5-s + (−0.959 + 0.281i)6-s + (0.654 − 0.755i)8-s + (−0.5 − 0.866i)9-s + (−0.888 + 0.458i)10-s + (0.888 + 0.458i)12-s + (0.841 − 0.540i)13-s + (−0.654 − 0.755i)15-s + (−0.995 + 0.0950i)16-s + (−0.786 − 0.618i)17-s + (−0.235 + 0.971i)18-s + (−0.786 + 0.618i)19-s + (0.959 + 0.281i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.876 + 0.482i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.876 + 0.482i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2209243354 - 0.8594311383i\)
\(L(\frac12)\) \(\approx\) \(-0.2209243354 - 0.8594311383i\)
\(L(1)\) \(\approx\) \(0.5248817748 - 0.6346622163i\)
\(L(1)\) \(\approx\) \(0.5248817748 - 0.6346622163i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.327 - 0.945i)T \)
13 \( 1 + (0.841 - 0.540i)T \)
17 \( 1 + (-0.786 - 0.618i)T \)
19 \( 1 + (-0.786 + 0.618i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (0.142 + 0.989i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (0.0475 - 0.998i)T \)
41 \( 1 + (-0.959 + 0.281i)T \)
43 \( 1 + (0.654 - 0.755i)T \)
47 \( 1 + (-0.235 - 0.971i)T \)
53 \( 1 + (-0.995 - 0.0950i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (0.235 + 0.971i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (0.327 - 0.945i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.52881645726268295177018890616, −21.794883550857887123011193477549, −20.97699907501727397127480463905, −20.03019772789659883808626582247, −19.21406731105612055480282677637, −18.67923148420451918831684609721, −17.56342160700457019042115865514, −17.116628507001797090966776559378, −15.832788699736663617710947559, −15.58318036867850234276790624284, −14.67037949816509149363982542532, −13.997615804721697625497071763097, −13.33930100185598904434499786475, −11.45748751930018843680357867652, −10.8461661220279135721995684893, −10.13362277574952545161895634584, −9.40366305125789846273371681400, −8.51016352343255679672500784567, −7.87019325365067039430018401284, −6.49663176681422655047120435744, −6.20635914665051362584993260358, −4.82148133415503557652463625634, −3.94984958520329403733975581879, −2.65330419829562681101447882697, −1.77542312723047050969480690327, 0.45769566967529277180295617632, 1.54928117117364030835755878504, 2.24445903828152695397171736881, 3.4030025883934833217377528129, 4.37423481941442549101179305299, 5.80264284861625968252803436298, 6.75835824456049360348679983164, 7.87311749231148097051753507116, 8.48346773766929406662841184156, 9.06688469608576968168326052681, 10.00494869606746593270051810697, 11.0253143750691963916767518641, 12.062525328682837710708015847787, 12.57423473461185838455806914181, 13.408312539310349813233464734159, 13.90549930719219895036239386961, 15.362502083959957494152704132627, 16.26258380339180798614733205368, 17.106349897559329276399266036635, 17.94965252583080229817703722320, 18.336052103989595061312461082200, 19.39759253644725636624231736164, 20.03512224894667457019398244809, 20.61812192039040279343746571281, 21.22499354552932138981565756568

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