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Properties

Label 1-891-891.5-r1-0-0
Degree $1$
Conductor $891$
Sign $0.680 + 0.733i$
Analytic cond. $95.7512$
Root an. cond. $95.7512$
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  + (0.459 − 0.888i)2-s + (−0.578 − 0.815i)4-s + (−0.795 + 0.606i)5-s + (−0.750 − 0.660i)7-s + (−0.990 + 0.139i)8-s + (0.173 + 0.984i)10-s + (0.352 − 0.935i)13-s + (−0.931 + 0.363i)14-s + (−0.331 + 0.943i)16-s + (−0.559 − 0.829i)17-s + (−0.615 − 0.788i)19-s + (0.954 + 0.297i)20-s + (−0.396 − 0.918i)23-s + (0.264 − 0.964i)25-s + (−0.669 − 0.743i)26-s + ⋯
L(s)  = 1  + (0.459 − 0.888i)2-s + (−0.578 − 0.815i)4-s + (−0.795 + 0.606i)5-s + (−0.750 − 0.660i)7-s + (−0.990 + 0.139i)8-s + (0.173 + 0.984i)10-s + (0.352 − 0.935i)13-s + (−0.931 + 0.363i)14-s + (−0.331 + 0.943i)16-s + (−0.559 − 0.829i)17-s + (−0.615 − 0.788i)19-s + (0.954 + 0.297i)20-s + (−0.396 − 0.918i)23-s + (0.264 − 0.964i)25-s + (−0.669 − 0.743i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.680 + 0.733i$
Analytic conductor: \(95.7512\)
Root analytic conductor: \(95.7512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 891,\ (1:\ ),\ 0.680 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2357416463 - 0.1028850795i\)
\(L(\frac12)\) \(\approx\) \(-0.2357416463 - 0.1028850795i\)
\(L(1)\) \(\approx\) \(0.5800863637 - 0.5328220513i\)
\(L(1)\) \(\approx\) \(0.5800863637 - 0.5328220513i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.459 - 0.888i)T \)
5 \( 1 + (-0.795 + 0.606i)T \)
7 \( 1 + (-0.750 - 0.660i)T \)
13 \( 1 + (0.352 - 0.935i)T \)
17 \( 1 + (-0.559 - 0.829i)T \)
19 \( 1 + (-0.615 - 0.788i)T \)
23 \( 1 + (-0.396 - 0.918i)T \)
29 \( 1 + (-0.931 - 0.363i)T \)
31 \( 1 + (0.871 - 0.489i)T \)
37 \( 1 + (-0.374 - 0.927i)T \)
41 \( 1 + (0.150 - 0.988i)T \)
43 \( 1 + (-0.686 + 0.727i)T \)
47 \( 1 + (0.417 + 0.908i)T \)
53 \( 1 + (-0.913 - 0.406i)T \)
59 \( 1 + (-0.0813 + 0.996i)T \)
61 \( 1 + (-0.0116 - 0.999i)T \)
67 \( 1 + (-0.0581 - 0.998i)T \)
71 \( 1 + (-0.438 + 0.898i)T \)
73 \( 1 + (-0.882 - 0.469i)T \)
79 \( 1 + (-0.459 + 0.888i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (0.795 + 0.606i)T \)
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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

−22.553400284797528284225366829343, −21.80602509963721390532270727188, −21.12654981184102399733013037451, −20.103005199106531404509434473053, −19.10131160237779472024790652811, −18.62229812967949569595175568818, −17.38501722814970642938164020343, −16.6641567936786533932450977421, −16.00898578114201653323263177336, −15.376663714964246284064673165510, −14.69393218349098367892223858942, −13.52275491792947834096930729285, −12.92112053092271062387587371895, −12.09568616144483942227288631724, −11.51914668424786564576224616602, −10.023872135860688349661145772862, −8.907176780508405726943206691881, −8.55528311160424531015697744563, −7.54522769432131607202295685387, −6.55083931689020122009301633854, −5.89307916866382573972459692264, −4.83058438263263996180220976313, −3.956749732111202053781762059301, −3.279737988633801349837333519238, −1.725756443346567030377201966782, 0.07337715629918448912797241797, 0.65613063834634029515453167282, 2.367327185153706447726130622866, 3.10804887335200843914242093901, 3.96160094307471116057941236166, 4.68677134570331645519011752305, 6.017042890023162689438578987960, 6.78924588113025913233379874578, 7.82128464651028004784697145558, 8.92079605380711999554762713792, 9.88450721777700612140856216645, 10.73284963924905548429177077050, 11.15357672929292648150963948616, 12.201981997393807217423334873326, 12.96665429795134861873694973731, 13.663360248655136197361164972672, 14.54208858503518722691098910829, 15.44196304721545911377021239553, 15.98497959034919787373335537105, 17.30153175727825823026878778982, 18.17263256074333930243211337323, 18.959354292805591040186687803154, 19.60668426919139378212375299497, 20.25876586958813159227620348345, 20.888510558783954743984313514987

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