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Properties

Label 1-847-847.837-r0-0-0
Degree $1$
Conductor $847$
Sign $0.998 + 0.0620i$
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  + (−0.995 − 0.0950i)2-s + (−0.5 − 0.866i)3-s + (0.981 + 0.189i)4-s + (0.235 − 0.971i)5-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.888 − 0.458i)17-s + (0.580 − 0.814i)18-s + (−0.888 + 0.458i)19-s + (0.415 − 0.909i)20-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (−0.5 − 0.866i)3-s + (0.981 + 0.189i)4-s + (0.235 − 0.971i)5-s + (0.415 + 0.909i)6-s + (−0.959 − 0.281i)8-s + (−0.5 + 0.866i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (−0.654 + 0.755i)13-s + (−0.959 + 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.888 − 0.458i)17-s + (0.580 − 0.814i)18-s + (−0.888 + 0.458i)19-s + (0.415 − 0.909i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.998 + 0.0620i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.998 + 0.0620i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5355206620 + 0.01663739366i\)
\(L(\frac12)\) \(\approx\) \(0.5355206620 + 0.01663739366i\)
\(L(1)\) \(\approx\) \(0.5286065858 - 0.1580645675i\)
\(L(1)\) \(\approx\) \(0.5286065858 - 0.1580645675i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.235 - 0.971i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (-0.888 + 0.458i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.841 + 0.540i)T \)
31 \( 1 + (-0.327 + 0.945i)T \)
37 \( 1 + (0.981 - 0.189i)T \)
41 \( 1 + (0.415 + 0.909i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.580 + 0.814i)T \)
53 \( 1 + (0.928 - 0.371i)T \)
59 \( 1 + (-0.995 + 0.0950i)T \)
61 \( 1 + (0.580 + 0.814i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (-0.786 + 0.618i)T \)
79 \( 1 + (0.235 - 0.971i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (-0.959 - 0.281i)T \)
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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

−21.87876497802564372116502767549, −21.44707638248278120969356206409, −20.38793084237059626983823024414, −19.68062856033450393723856879307, −18.79072079849569072332193417055, −17.936234163631800551725243112193, −17.28588356897782467386050412612, −16.80516162789629362498611243561, −15.51540848707287142239749962735, −15.19855829115453405800425352667, −14.524217045308694790949729771783, −13.08822654701662592841466937551, −11.939778922282405591797814501956, −11.04398989906382219587265018188, −10.606234066924720360248495258642, −9.88776337760188397326001591388, −9.07557471313943312009213233606, −8.120658939770581944940651601062, −6.96844243500676935623757371106, −6.3419040497855473774141237460, −5.45620605772725579998005850166, −4.218214168576476206457632043462, −2.96962394459480649089112707955, −2.245850021526423651087347082454, −0.43085783567011652794757305015, 0.95706431683791001027030981604, 1.82202938815002544534909984221, 2.69704756458643645941623104150, 4.45066903969528325950894883964, 5.430073324771657639984676447014, 6.497899006802276318254499190138, 7.09348725032819503681824169460, 8.13937428496593919699779394571, 8.82643915747370004744774388938, 9.61405151534417527949025868444, 10.72710211915319740624333962097, 11.50163087959395311427577092651, 12.31155665260050215587671193137, 12.86982663222571949219724948090, 13.87252157030352645351996646950, 15.02015855274589368213961474154, 16.2509320198457778271657156830, 16.63617365222408739937478258695, 17.44068682879837570958375708405, 17.98381957341576866050738694747, 18.919519464144892183057339841569, 19.647047471196837005113800013416, 20.16024934861180617785788054640, 21.312761367277231610799964471263, 21.81756809311725622966900029678

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