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Properties

Label 1-33e2-1089.344-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.190 + 0.981i$
Analytic cond. $117.029$
Root an. cond. $117.029$
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  + (−0.879 + 0.475i)2-s + (0.548 − 0.836i)4-s + (−0.483 − 0.875i)5-s + (−0.948 + 0.318i)7-s + (−0.0855 + 0.996i)8-s + (0.841 + 0.540i)10-s + (0.964 + 0.263i)13-s + (0.683 − 0.730i)14-s + (−0.398 − 0.917i)16-s + (0.985 − 0.170i)17-s + (0.696 + 0.717i)19-s + (−0.997 − 0.0760i)20-s + (−0.580 − 0.814i)23-s + (−0.532 + 0.846i)25-s + (−0.974 + 0.226i)26-s + ⋯
L(s)  = 1  + (−0.879 + 0.475i)2-s + (0.548 − 0.836i)4-s + (−0.483 − 0.875i)5-s + (−0.948 + 0.318i)7-s + (−0.0855 + 0.996i)8-s + (0.841 + 0.540i)10-s + (0.964 + 0.263i)13-s + (0.683 − 0.730i)14-s + (−0.398 − 0.917i)16-s + (0.985 − 0.170i)17-s + (0.696 + 0.717i)19-s + (−0.997 − 0.0760i)20-s + (−0.580 − 0.814i)23-s + (−0.532 + 0.846i)25-s + (−0.974 + 0.226i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.190 + 0.981i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.190 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3866088320 + 0.4687873257i\)
\(L(\frac12)\) \(\approx\) \(0.3866088320 + 0.4687873257i\)
\(L(1)\) \(\approx\) \(0.5916546132 + 0.06573911539i\)
\(L(1)\) \(\approx\) \(0.5916546132 + 0.06573911539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.879 + 0.475i)T \)
5 \( 1 + (-0.483 - 0.875i)T \)
7 \( 1 + (-0.948 + 0.318i)T \)
13 \( 1 + (0.964 + 0.263i)T \)
17 \( 1 + (0.985 - 0.170i)T \)
19 \( 1 + (0.696 + 0.717i)T \)
23 \( 1 + (-0.580 - 0.814i)T \)
29 \( 1 + (-0.999 + 0.0380i)T \)
31 \( 1 + (0.161 + 0.986i)T \)
37 \( 1 + (-0.362 - 0.931i)T \)
41 \( 1 + (-0.380 - 0.924i)T \)
43 \( 1 + (0.981 + 0.189i)T \)
47 \( 1 + (0.761 + 0.647i)T \)
53 \( 1 + (-0.993 + 0.113i)T \)
59 \( 1 + (0.991 - 0.132i)T \)
61 \( 1 + (-0.851 + 0.524i)T \)
67 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (-0.897 + 0.441i)T \)
73 \( 1 + (-0.870 + 0.491i)T \)
79 \( 1 + (0.797 + 0.603i)T \)
83 \( 1 + (-0.953 - 0.299i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.483 - 0.875i)T \)
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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

−20.70803365949036980079186590264, −20.18089737533711173815866438444, −19.25386928234083729682525558602, −18.847756728342316953846299563321, −18.12206571952172634714608280535, −17.26727289060849053884397012906, −16.350172999530432641217256995591, −15.755239375292004462097229312166, −15.04309605257360908434192298731, −13.73238091854093552172552656846, −13.116252379385908381265489086037, −12.0232941837516359115513140470, −11.40416517493753913819613967723, −10.58907167145055278208968408004, −9.9047396423846413026618177570, −9.201479098373239017766517759923, −8.01210966640111908254360926975, −7.46336310543332837141966701009, −6.6000325567905253359646023748, −5.78664018094894430314969524257, −3.96706510707958629617943384043, −3.40752681581409739729398709615, −2.68457186672679147154057405714, −1.32010516717739936933410172181, −0.23792834460655656622019482846, 0.79167154079269729847793266101, 1.71670782314391248439937712884, 3.11991579629086501561292174575, 4.106962925880889727483739609512, 5.48461721819883224740356581234, 5.91844137360009442398429429115, 7.05032424987262300162957265040, 7.84217147083277748480183319163, 8.7023304894165989124290135973, 9.265825348784245013288675544725, 10.08366946920897641335487819523, 10.99306701307440446965848796759, 12.05338486978414773065604977854, 12.54510795883113401954399090291, 13.7457474616331453239806034763, 14.53141671352375527487616324808, 15.729430531138615420451438094606, 16.071727850358787682306789852, 16.57336300894682721250492899196, 17.49191052440442836944939661135, 18.586398352769093703249619097531, 18.91433578334930665377452880484, 19.79506536248834459234479769724, 20.55161809641026102567897456893, 21.07717425556531385490512514178

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