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Properties

Label 1-3e4-81.25-r0-0-0
Degree $1$
Conductor $81$
Sign $-0.565 - 0.824i$
Analytic cond. $0.376162$
Root an. cond. $0.376162$
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  + (0.597 − 0.802i)2-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.893 + 0.448i)11-s + (0.396 − 0.918i)13-s + (−0.993 + 0.116i)14-s + (−0.835 + 0.549i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 + 0.957i)20-s + (0.893 − 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯
L(s)  = 1  + (0.597 − 0.802i)2-s + (−0.286 − 0.957i)4-s + (−0.835 − 0.549i)5-s + (−0.686 − 0.727i)7-s + (−0.939 − 0.342i)8-s + (−0.939 + 0.342i)10-s + (0.893 + 0.448i)11-s + (0.396 − 0.918i)13-s + (−0.993 + 0.116i)14-s + (−0.835 + 0.549i)16-s + (0.766 + 0.642i)17-s + (0.766 − 0.642i)19-s + (−0.286 + 0.957i)20-s + (0.893 − 0.448i)22-s + (−0.686 + 0.727i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(81\)    =    \(3^{4}\)
Sign: $-0.565 - 0.824i$
Analytic conductor: \(0.376162\)
Root analytic conductor: \(0.376162\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{81} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 81,\ (0:\ ),\ -0.565 - 0.824i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4826478108 - 0.9162842663i\)
\(L(\frac12)\) \(\approx\) \(0.4826478108 - 0.9162842663i\)
\(L(1)\) \(\approx\) \(0.8578892379 - 0.7207410342i\)
\(L(1)\) \(\approx\) \(0.8578892379 - 0.7207410342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
good2 \( 1 + (0.597 - 0.802i)T \)
5 \( 1 + (-0.835 - 0.549i)T \)
7 \( 1 + (-0.686 - 0.727i)T \)
11 \( 1 + (0.893 + 0.448i)T \)
13 \( 1 + (0.396 - 0.918i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.686 + 0.727i)T \)
29 \( 1 + (-0.993 - 0.116i)T \)
31 \( 1 + (0.973 + 0.230i)T \)
37 \( 1 + (0.173 - 0.984i)T \)
41 \( 1 + (0.597 + 0.802i)T \)
43 \( 1 + (-0.0581 - 0.998i)T \)
47 \( 1 + (0.973 - 0.230i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.893 - 0.448i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (-0.993 + 0.116i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (0.597 - 0.802i)T \)
83 \( 1 + (0.597 - 0.802i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.835 + 0.549i)T \)
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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

−31.529969382680999141835040836873, −30.61886324473565805717498373058, −29.54402908794323927457068197536, −27.968281232716587408494634485788, −26.8467174952329137781941871021, −25.962538836541524338886622714006, −24.87022982538840632287495775116, −23.87523795283431177074801697603, −22.63842288302951035654969962929, −22.203702134077361894041331788167, −20.770075475387497822753640424769, −19.139947646283328146047999569449, −18.354992261008573410364985159802, −16.56882151221462062886526320164, −15.97083311149643414910913605713, −14.740545401410681514594137991546, −13.86169700705260880860854258216, −12.23246317873356635297210428624, −11.5787876110271670964739964773, −9.45386826804587927242220334923, −8.18906693073036944198462769328, −6.87027268007125274325310199269, −5.89569063270494279604925241259, −4.11463248023534037973483374007, −3.04450904163407750102405370090, 1.0818412953131287762149612073, 3.35749710995763315756379212740, 4.22309363683394877779956355808, 5.79800777997070160672523305999, 7.46563314459582469929018112729, 9.17870438646454998304589557674, 10.34906793113130124209643173893, 11.655294419898429544911897041762, 12.60945444162282599741548495721, 13.61499647094199702322243009472, 15.00197310359883742722795300261, 16.088380540080799727849770277043, 17.5249873734496733431229487598, 19.16209853167684382869557633600, 19.909483634010842985243713340792, 20.60587461249004821096378759863, 22.15552574880729659615316446549, 23.02909675410983724770221974557, 23.81108285757371470804134192804, 25.07760406719390940907678064919, 26.652339806980353309065369961082, 27.83911599468915298663153478143, 28.40337615694487148605368064359, 29.94324341847735918850478998785, 30.37834859710089003017769799879

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