Properties

Label 1-847-847.683-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.152 + 0.988i$
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  + (0.928 + 0.371i)2-s + (−0.5 − 0.866i)3-s + (0.723 + 0.690i)4-s + (0.580 + 0.814i)5-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.235 + 0.971i)10-s + (0.235 − 0.971i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (0.0475 + 0.998i)16-s + (−0.327 + 0.945i)17-s + (−0.786 + 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.142 + 0.989i)20-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)2-s + (−0.5 − 0.866i)3-s + (0.723 + 0.690i)4-s + (0.580 + 0.814i)5-s + (−0.142 − 0.989i)6-s + (0.415 + 0.909i)8-s + (−0.5 + 0.866i)9-s + (0.235 + 0.971i)10-s + (0.235 − 0.971i)12-s + (−0.959 + 0.281i)13-s + (0.415 − 0.909i)15-s + (0.0475 + 0.998i)16-s + (−0.327 + 0.945i)17-s + (−0.786 + 0.618i)18-s + (−0.327 − 0.945i)19-s + (−0.142 + 0.989i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.152 + 0.988i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (683, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.152 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289763821 + 1.503878369i\)
\(L(\frac12)\) \(\approx\) \(1.289763821 + 1.503878369i\)
\(L(1)\) \(\approx\) \(1.436262463 + 0.5258364063i\)
\(L(1)\) \(\approx\) \(1.436262463 + 0.5258364063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.928 + 0.371i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.580 + 0.814i)T \)
13 \( 1 + (-0.959 + 0.281i)T \)
17 \( 1 + (-0.327 + 0.945i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
23 \( 1 + (0.0475 + 0.998i)T \)
29 \( 1 + (-0.654 + 0.755i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (0.723 - 0.690i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (0.0475 - 0.998i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (-0.786 - 0.618i)T \)
67 \( 1 + (-0.786 + 0.618i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (-0.888 - 0.458i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (0.841 + 0.540i)T \)
89 \( 1 + (0.981 - 0.189i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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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

−22.027347183614063634402106669340, −21.01147206743070095722241237061, −20.63238974829222852485958995859, −19.99245516466874484843333134621, −18.8165172255722747929463331137, −17.74074407635838622968714859009, −16.668138128950815385953431547586, −16.44722253806017806399553566400, −15.274935702552100803315638661784, −14.70794296369063097593668783121, −13.756010534835505098016712039817, −12.88145674828041863884164257733, −12.10235752372270316834621527672, −11.47347100306594618917284653879, −10.33490873808311920536673389818, −9.84530736533471261375032456109, −9.01281485364133588324329939285, −7.635020102674109126400013244629, −6.25844196876101783840972802390, −5.75175359366049095495509085542, −4.65833744925140016068004666092, −4.44515976838266744915916170877, −3.05989650608684281721663272510, −2.09349734107331631790030032881, −0.64035815818634513730926521344, 1.748669621305373657677931315933, 2.42486429254896874329909242585, 3.47257708610254717007430689657, 4.82442029560709596205340569321, 5.59238247789526809536456817904, 6.46428686133968975232798513356, 7.04188923024180641934725187011, 7.73901673363517860295872191209, 8.981461994966293057569473850314, 10.37746895965317917412909846320, 11.138213883433582398178591099613, 11.832742950621721972872378720326, 12.890136506274256680247526669509, 13.26822016214811134059303090261, 14.310681826370446138226493091185, 14.76139992456953708806306636183, 15.81885549069481429162027907973, 16.87594406847922598233886309411, 17.50038422110013394308815185106, 18.03625496954674413681220151615, 19.38737195074645571648252534921, 19.66371642094149053576723431599, 21.129513502605077911000845999684, 21.92392031789710966020485267686, 22.20419774446822350798112029017

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