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Properties

Label 1-847-847.639-r0-0-0
Degree $1$
Conductor $847$
Sign $0.208 - 0.977i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.208 - 0.977i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.208 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3627660583 + 0.2934431678i\)
\(L(\frac12)\) \(\approx\) \(-0.3627660583 + 0.2934431678i\)
\(L(1)\) \(\approx\) \(0.5506098774 + 0.7284523214i\)
\(L(1)\) \(\approx\) \(0.5506098774 + 0.7284523214i\)

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

−21.75535558872620893984182820525, −20.43696227065835844618928324475, −20.09682768636940483091513637160, −19.164634373207438474862152809269, −18.543086148409832910963213521513, −17.593789171604550760055096291159, −16.66300546018244171148800444978, −15.86223128987510543375915966577, −14.92144470154086512877010281914, −13.75372810002669651184957966981, −13.2142106644731529713773900336, −12.60088463010475380123709274553, −11.77326637630825377323469740028, −11.19405258704324997451587062741, −10.30939125900786992032672881875, −8.94752477500520592324977594272, −8.36837596405503547703023255178, −6.9757615462954842014326502363, −6.22788997083186577028380357753, −5.33570609974219999002061332922, −4.48068746212639870114477075590, −3.60347468319733111983778637897, −2.074838236247469604352623587911, −1.50674306081671602463453144783, −0.16575609573739327285939783935, 2.37030488286487035646061463413, 3.513564973258800439303431759210, 3.98717147204427420535989964382, 5.034198205595824362526592091903, 6.084880395166778733348099516687, 6.491606237042331829257336052370, 7.68069159663317170790783766599, 8.51311981388871896696204656467, 9.634106034958226408785084295660, 10.71790738521883375592335102535, 11.28751932677829683538404072686, 12.104605462479891670524525472859, 13.13726532982982081839644205557, 14.15801359961172776061236927735, 14.76512507500869061897316368242, 15.64321092864501596931023136127, 15.94901805125297629969490770722, 16.9439412942769127407156815317, 17.86851554497220835101761265048, 18.34528990160181621049454624315, 19.77480826355630521445539036058, 20.658968937867908809696136613314, 21.39337905067126050597628811553, 22.189033637335215261195296503097, 22.694789771489287528833343832003

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