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Properties

Label 1-33e2-1089.256-r0-0-0
Degree $1$
Conductor $1089$
Sign $0.999 + 0.00576i$
Analytic cond. $5.05729$
Root an. cond. $5.05729$
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  + (−0.830 + 0.556i)2-s + (0.380 − 0.924i)4-s + (0.953 + 0.299i)5-s + (0.548 − 0.836i)7-s + (0.198 + 0.980i)8-s + (−0.959 + 0.281i)10-s + (0.997 + 0.0760i)13-s + (0.00951 + 0.999i)14-s + (−0.710 − 0.703i)16-s + (−0.921 − 0.389i)17-s + (0.974 + 0.226i)19-s + (0.640 − 0.768i)20-s + (0.0475 − 0.998i)23-s + (0.820 + 0.572i)25-s + (−0.870 + 0.491i)26-s + ⋯
L(s)  = 1  + (−0.830 + 0.556i)2-s + (0.380 − 0.924i)4-s + (0.953 + 0.299i)5-s + (0.548 − 0.836i)7-s + (0.198 + 0.980i)8-s + (−0.959 + 0.281i)10-s + (0.997 + 0.0760i)13-s + (0.00951 + 0.999i)14-s + (−0.710 − 0.703i)16-s + (−0.921 − 0.389i)17-s + (0.974 + 0.226i)19-s + (0.640 − 0.768i)20-s + (0.0475 − 0.998i)23-s + (0.820 + 0.572i)25-s + (−0.870 + 0.491i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00576i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.999 + 0.00576i$
Analytic conductor: \(5.05729\)
Root analytic conductor: \(5.05729\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (0:\ ),\ 0.999 + 0.00576i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.398003437 + 0.004033029854i\)
\(L(\frac12)\) \(\approx\) \(1.398003437 + 0.004033029854i\)
\(L(1)\) \(\approx\) \(0.9845060987 + 0.1002223675i\)
\(L(1)\) \(\approx\) \(0.9845060987 + 0.1002223675i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.830 + 0.556i)T \)
5 \( 1 + (0.953 + 0.299i)T \)
7 \( 1 + (0.548 - 0.836i)T \)
13 \( 1 + (0.997 + 0.0760i)T \)
17 \( 1 + (-0.921 - 0.389i)T \)
19 \( 1 + (0.974 + 0.226i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (-0.905 - 0.424i)T \)
31 \( 1 + (0.879 - 0.475i)T \)
37 \( 1 + (0.941 + 0.336i)T \)
41 \( 1 + (-0.532 + 0.846i)T \)
43 \( 1 + (0.580 + 0.814i)T \)
47 \( 1 + (-0.969 + 0.244i)T \)
53 \( 1 + (-0.254 - 0.967i)T \)
59 \( 1 + (0.999 - 0.0380i)T \)
61 \( 1 + (-0.0665 + 0.997i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (0.516 - 0.856i)T \)
73 \( 1 + (-0.362 - 0.931i)T \)
79 \( 1 + (-0.398 + 0.917i)T \)
83 \( 1 + (-0.935 + 0.353i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.953 - 0.299i)T \)
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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

−21.47733966528182157013613375468, −20.53370449005830129119238565673, −20.07821275471037041543667613506, −18.887293884340230349172864476888, −18.31345331289932536730640704251, −17.62977812268228056949008044376, −17.17116196743980102232559159447, −15.96700776294244535830803149093, −15.53813202194705125686882272894, −14.249353712159253018726310043451, −13.33954815378348149115699004593, −12.76043780484425504200419238917, −11.68785723685359799005693764493, −11.14192374482780589470075708896, −10.22234967681506498883930320693, −9.28860105437795030467394423158, −8.83743670772862299219388629110, −8.050395321916797953765419785332, −6.93421682473809963422480356664, −5.95519050404737772173738542558, −5.13612664469326699910593340983, −3.87293881662763090543651443926, −2.76659879715374868442334623068, −1.89157942465929644054429825727, −1.16722116619905501816520174230, 0.90872513846334722787650580296, 1.76254065136073425911589716649, 2.824083267901045467078973295094, 4.31035491763730608723027588758, 5.249672638994554648735984196988, 6.25320727047027934859302146858, 6.77067463210650413308603013333, 7.79009106198451704335081415439, 8.51859517105340789697872710563, 9.54661848001562291769763736638, 10.054381757601342927372798321563, 11.08468025167757070360797215182, 11.38940693040587521468163643836, 13.15042935336369225704374362219, 13.722286911449183709846961536812, 14.44239009541243822179753152449, 15.196129675256766940039254796342, 16.287836775269849281413120228225, 16.75674322371510367330080133640, 17.760491890275529854259242244177, 18.07056949060620562531775603989, 18.809835903634170630523461494120, 19.91958225656934074807141368530, 20.62761135739335783572356249000, 21.087782856790881607426493578822

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