↑

Properties

Label 1-861-861.566-r0-0-0
Degree $1$
Conductor $861$
Sign $0.920 - 0.389i$
Analytic cond. $3.99846$
Root an. cond. $3.99846$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.809 + 0.587i)10-s + (0.587 − 0.809i)11-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)19-s + (−0.309 + 0.951i)20-s + (0.951 + 0.309i)22-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.587 − 0.809i)26-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.809 − 0.587i)8-s + (0.809 + 0.587i)10-s + (0.587 − 0.809i)11-s + (−0.951 + 0.309i)13-s + (0.309 − 0.951i)16-s + (0.587 − 0.809i)17-s + (−0.951 − 0.309i)19-s + (−0.309 + 0.951i)20-s + (0.951 + 0.309i)22-s + (−0.309 − 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.587 − 0.809i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $0.920 - 0.389i$
Analytic conductor: \(3.99846\)
Root analytic conductor: \(3.99846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (0:\ ),\ 0.920 - 0.389i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.397124209 - 0.2836603814i\)
\(L(\frac12)\) \(\approx\) \(1.397124209 - 0.2836603814i\)
\(L(1)\) \(\approx\) \(1.149863619 + 0.2291872873i\)
\(L(1)\) \(\approx\) \(1.149863619 + 0.2291872873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (-0.309 - 0.951i)T \)
29 \( 1 + (-0.587 - 0.809i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.951 + 0.309i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + (0.587 - 0.809i)T \)
73 \( 1 + T \)
79 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 + (-0.951 - 0.309i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.007373872210731730724396834567, −21.385439955212258293830201363417, −20.665269528644502057987353802267, −19.63163536659285631899681351753, −19.19863266461915303278960002381, −18.182264900377246194870019033915, −17.46078208076393665012949366645, −16.939658969614021406209854472508, −15.23477461686490274047143957293, −14.67871365115322250478257230420, −14.09358235976679493908009028310, −13.00338630268463250305871698800, −12.48219583852674412518183491971, −11.53974789777196479607351861112, −10.59062660248212808189840833427, −9.90829781033593359257889612807, −9.40474826106695567556219144178, −8.182093951516868756738401930115, −6.98522303138968445742190160674, −6.003967390571710381639154901782, −5.19800067014112758502836207347, −4.135365432715331189935063539281, −3.200762260224518670789155297133, −2.14618249677474585710311187244, −1.503741245526258025140025396034, 0.56695810087788272617635179699, 2.1487475748774048305513302099, 3.35111834507381952066563570944, 4.57531160646000790890522010919, 5.13537346382987185850378298816, 6.2041026375196830414534215156, 6.73460551693079551830849313933, 7.95244276895498163488994329012, 8.7400082054390148876815061377, 9.44710805679350424322327790537, 10.279254880891503955393893208100, 11.7629317777682520113304692932, 12.43020434729810047204958167690, 13.36128071091410512874286204478, 14.042799742595812437706167115849, 14.61749709600959946590636865801, 15.669979180801593459588076424761, 16.66669888317852815956278013594, 16.912585390076281703618408578436, 17.74139647577905239883410584684, 18.704115209240134272584388210820, 19.48663320937792393771738721555, 20.77744609061949631472162858987, 21.31985215978743364259591673147, 22.14964317930572345589930138326

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