↑

Properties

Label 1-861-861.296-r1-0-0
Degree $1$
Conductor $861$
Sign $-0.920 - 0.390i$
Analytic cond. $92.5273$
Root an. cond. $92.5273$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + 20-s − i·22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.866 − 0.5i)26-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 8-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + i·13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.5i)19-s + 20-s − i·22-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.866 − 0.5i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(92.5273\)
Root analytic conductor: \(92.5273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (1:\ ),\ -0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1338275010 - 0.6589299509i\)
\(L(\frac12)\) \(\approx\) \(0.1338275010 - 0.6589299509i\)
\(L(1)\) \(\approx\) \(0.6113090153 - 0.2912378658i\)
\(L(1)\) \(\approx\) \(0.6113090153 - 0.2912378658i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.866 - 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 - iT \)
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

−22.34103106390285681253078757202, −21.82088949920953540644437089787, −20.242946111805829699486896781908, −19.65254796172427283200467303875, −18.88342669974135072932711155945, −18.23282378053642372633264694229, −17.34663171423080578847672277357, −16.72634716845998159363700447312, −15.66164898816047538263701282312, −15.05112860911656491663900631255, −14.52694175075139776069296667851, −13.53751499468738781136032581851, −12.6265853646679761862418141697, −11.23587118359367044206249568811, −10.774774240211270291653059106225, −9.88363953935165315728815361624, −8.678408744127808064148598183006, −8.2845305884720019763408961979, −7.02730218716847556834458035303, −6.625762722694615307884189424671, −5.674552530809032500330268276554, −4.49386940892720817432021615547, −3.571239514070294631427826435519, −2.29483982025848302630572335294, −0.828741814617035126726551391942, 0.23489940994913585520429002871, 1.44413728884196645890591658411, 2.18452724669996696899718852, 3.72801030148035867480744031033, 4.223348880427491588361247518385, 5.15375114711599055779668061172, 6.718469654569699027198511211383, 7.5200521666378381328207942507, 8.68982981392871173722051377363, 9.051322980328982521056462162393, 9.9055554481166637470808673363, 11.07527087513163238440726921736, 11.751106750110746824972034347083, 12.33575755982497563910719034771, 13.21591027575726720808862410704, 14.0100138450518451317894746082, 15.18019057191605587478843920528, 16.17538375333779289323824154142, 16.92221487785454164170998355861, 17.46233321135867193158683869436, 18.48614406521065090226687028860, 19.50615482895115169135574681421, 19.674691992756072414896832121047, 20.70816115893868239015406210858, 21.27015098582061746493691502890

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