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Properties

Label 1-33e2-1089.283-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.944 - 0.329i$
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  + (−0.964 − 0.263i)2-s + (0.861 + 0.508i)4-s + (0.640 + 0.768i)5-s + (0.290 − 0.956i)7-s + (−0.696 − 0.717i)8-s + (−0.415 − 0.909i)10-s + (0.217 − 0.976i)13-s + (−0.532 + 0.846i)14-s + (0.483 + 0.875i)16-s + (0.0285 + 0.999i)17-s + (−0.610 + 0.791i)19-s + (0.161 + 0.986i)20-s + (−0.327 + 0.945i)23-s + (−0.179 + 0.983i)25-s + (−0.466 + 0.884i)26-s + ⋯
L(s)  = 1  + (−0.964 − 0.263i)2-s + (0.861 + 0.508i)4-s + (0.640 + 0.768i)5-s + (0.290 − 0.956i)7-s + (−0.696 − 0.717i)8-s + (−0.415 − 0.909i)10-s + (0.217 − 0.976i)13-s + (−0.532 + 0.846i)14-s + (0.483 + 0.875i)16-s + (0.0285 + 0.999i)17-s + (−0.610 + 0.791i)19-s + (0.161 + 0.986i)20-s + (−0.327 + 0.945i)23-s + (−0.179 + 0.983i)25-s + (−0.466 + 0.884i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05851088061 - 0.3455127635i\)
\(L(\frac12)\) \(\approx\) \(0.05851088061 - 0.3455127635i\)
\(L(1)\) \(\approx\) \(0.7037019575 - 0.07250305165i\)
\(L(1)\) \(\approx\) \(0.7037019575 - 0.07250305165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.964 - 0.263i)T \)
5 \( 1 + (0.640 + 0.768i)T \)
7 \( 1 + (0.290 - 0.956i)T \)
13 \( 1 + (0.217 - 0.976i)T \)
17 \( 1 + (0.0285 + 0.999i)T \)
19 \( 1 + (-0.610 + 0.791i)T \)
23 \( 1 + (-0.327 + 0.945i)T \)
29 \( 1 + (0.761 - 0.647i)T \)
31 \( 1 + (-0.595 - 0.803i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (-0.988 + 0.151i)T \)
43 \( 1 + (-0.928 - 0.371i)T \)
47 \( 1 + (0.449 - 0.893i)T \)
53 \( 1 + (0.516 + 0.856i)T \)
59 \( 1 + (-0.625 + 0.780i)T \)
61 \( 1 + (0.710 + 0.703i)T \)
67 \( 1 + (-0.888 - 0.458i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (-0.0855 - 0.996i)T \)
79 \( 1 + (0.830 - 0.556i)T \)
83 \( 1 + (-0.797 + 0.603i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.640 - 0.768i)T \)
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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

−21.47209960497501889211892811054, −20.6721618419349444575583847578, −20.05873799022791201696653061674, −19.06178771342411432253503289183, −18.357002666905159951772864445809, −17.80945301611309054322016685226, −16.92938580600158791375616797741, −16.2442460415862886221101039415, −15.65650111765097937825223215090, −14.60849447052553136490853257969, −13.94219686372800531600483645596, −12.771166074134032005417575603486, −11.93704451639670598982509969596, −11.28236465733653733226781584927, −10.19543192901803501286449619703, −9.39980736512174464422619287280, −8.719753615641403809781197031624, −8.32967336551820649274099621625, −6.90489631439791465229828960333, −6.34955305006012056001152123185, −5.269868552737545972358725410989, −4.67127187630980967483906120651, −2.843424181424003348589867158759, −2.03656610946682028994355577264, −1.18631990016842353705901757289, 0.09711020251779569795263440363, 1.36273119494551083023582122020, 2.11920010749929813214873949067, 3.30021897149735486620228172484, 3.992292509647919344575667290818, 5.68168819055112199909369897833, 6.334041478485462418444822431300, 7.37189072210019708412719612921, 7.91317376784340910220298090699, 8.86704828596666813914952106759, 10.12370312212309688523307764169, 10.28184721491623474114813299104, 11.02572586430978258564521234370, 11.97498793641812847425125569291, 13.054693016865253805949649696139, 13.712416035643052076079084661609, 14.87091471971923198006175663450, 15.337307819528706511425786507045, 16.64327621231991492276498373389, 17.117766351521276811113554581569, 17.86322383344530869865869562213, 18.397966209951794897659795261682, 19.39556480455983516398915426744, 19.94726610509420117936726543445, 20.878187854421707838021582903926

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