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Properties

Label 1-847-847.626-r0-0-0
Degree $1$
Conductor $847$
Sign $0.538 - 0.842i$
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  + (0.327 + 0.945i)2-s + (0.5 + 0.866i)3-s + (−0.786 + 0.618i)4-s + (−0.0475 − 0.998i)5-s + (−0.654 + 0.755i)6-s + (−0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (0.928 − 0.371i)10-s + (−0.928 − 0.371i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.995 + 0.0950i)17-s + (−0.981 − 0.189i)18-s + (−0.995 − 0.0950i)19-s + (0.654 + 0.755i)20-s + ⋯
L(s)  = 1  + (0.327 + 0.945i)2-s + (0.5 + 0.866i)3-s + (−0.786 + 0.618i)4-s + (−0.0475 − 0.998i)5-s + (−0.654 + 0.755i)6-s + (−0.841 − 0.540i)8-s + (−0.5 + 0.866i)9-s + (0.928 − 0.371i)10-s + (−0.928 − 0.371i)12-s + (−0.142 − 0.989i)13-s + (0.841 − 0.540i)15-s + (0.235 − 0.971i)16-s + (−0.995 + 0.0950i)17-s + (−0.981 − 0.189i)18-s + (−0.995 − 0.0950i)19-s + (0.654 + 0.755i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.538 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.538 - 0.842i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.538 - 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4128096575 - 0.2262337366i\)
\(L(\frac12)\) \(\approx\) \(0.4128096575 - 0.2262337366i\)
\(L(1)\) \(\approx\) \(0.8220324200 + 0.4399204688i\)
\(L(1)\) \(\approx\) \(0.8220324200 + 0.4399204688i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.327 + 0.945i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.0475 - 0.998i)T \)
13 \( 1 + (-0.142 - 0.989i)T \)
17 \( 1 + (-0.995 + 0.0950i)T \)
19 \( 1 + (-0.995 - 0.0950i)T \)
23 \( 1 + (0.235 - 0.971i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (-0.786 - 0.618i)T \)
41 \( 1 + (-0.654 + 0.755i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 + (-0.981 + 0.189i)T \)
53 \( 1 + (0.235 + 0.971i)T \)
59 \( 1 + (0.327 - 0.945i)T \)
61 \( 1 + (0.981 - 0.189i)T \)
67 \( 1 + (0.981 + 0.189i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.723 + 0.690i)T \)
79 \( 1 + (-0.0475 - 0.998i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.580 - 0.814i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−22.15408189373745197566579504644, −21.44711142367200664653015311844, −20.59407733854523599100947196701, −19.53232907799026675561974141458, −19.335570696114765095607594037711, −18.38683832761298755519312322718, −17.951100134389108438904136818025, −16.928579302477997010772146130651, −15.29809050205947280173626913738, −14.76307752590075898705179768936, −13.959772477057241810947648397734, −13.35616959523973632752275837062, −12.53991051357145414538244436235, −11.51755639159470796527974733851, −11.128405918798352310682475644803, −9.978322563278695017895916678052, −9.10937930039027909366935752613, −8.310918294512480411273153859647, −7.001861961765747447492253637844, −6.53212399534733255093082719653, −5.3172121859659492356303552930, −3.96942148176806527083781814964, −3.28924841730806405763389005319, −2.19123833715587503833765571233, −1.71510003224518210526276548401, 0.16195548884749587876664191246, 2.21683282796027340923637302235, 3.48624927309474471595552482663, 4.32714728822653541904537562666, 4.99561952954487133288366865572, 5.771800610124011007361290184782, 6.94864569376012641294242601426, 8.2304480665698363895846008097, 8.47668158841396489617045635734, 9.3603059419092102186851758849, 10.24441083598257165999812831914, 11.36671990503886724820620743698, 12.725505271018445764292704709167, 13.06186937416932947409640909943, 14.06767854204680830525755390631, 14.977858335303280239020974129184, 15.5003596846446684631897386024, 16.23223116872275356412831648938, 17.01307607478422338095475085146, 17.53456516708483292536389042363, 18.765114518616088193959499545785, 19.8714007574429608313387349279, 20.46899857923737911513364215866, 21.31584011896348011174357718158, 21.97597760692815350872419394918

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