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Properties

Label 1-33e2-1089.151-r1-0-0
Degree $1$
Conductor $1089$
Sign $0.292 + 0.956i$
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  + (0.595 + 0.803i)2-s + (−0.290 + 0.956i)4-s + (0.997 − 0.0760i)5-s + (0.969 + 0.244i)7-s + (−0.941 + 0.336i)8-s + (0.654 + 0.755i)10-s + (0.999 − 0.0190i)13-s + (0.380 + 0.924i)14-s + (−0.830 − 0.556i)16-s + (−0.774 − 0.633i)17-s + (0.998 − 0.0570i)19-s + (−0.217 + 0.976i)20-s + (0.928 + 0.371i)23-s + (0.988 − 0.151i)25-s + (0.610 + 0.791i)26-s + ⋯
L(s)  = 1  + (0.595 + 0.803i)2-s + (−0.290 + 0.956i)4-s + (0.997 − 0.0760i)5-s + (0.969 + 0.244i)7-s + (−0.941 + 0.336i)8-s + (0.654 + 0.755i)10-s + (0.999 − 0.0190i)13-s + (0.380 + 0.924i)14-s + (−0.830 − 0.556i)16-s + (−0.774 − 0.633i)17-s + (0.998 − 0.0570i)19-s + (−0.217 + 0.976i)20-s + (0.928 + 0.371i)23-s + (0.988 − 0.151i)25-s + (0.610 + 0.791i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $0.292 + 0.956i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ 0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.795619352 + 2.809034229i\)
\(L(\frac12)\) \(\approx\) \(3.795619352 + 2.809034229i\)
\(L(1)\) \(\approx\) \(1.807711945 + 0.9387343470i\)
\(L(1)\) \(\approx\) \(1.807711945 + 0.9387343470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.595 + 0.803i)T \)
5 \( 1 + (0.997 - 0.0760i)T \)
7 \( 1 + (0.969 + 0.244i)T \)
13 \( 1 + (0.999 - 0.0190i)T \)
17 \( 1 + (-0.774 - 0.633i)T \)
19 \( 1 + (0.998 - 0.0570i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (0.625 - 0.780i)T \)
31 \( 1 + (0.123 - 0.992i)T \)
37 \( 1 + (0.0855 + 0.996i)T \)
41 \( 1 + (-0.861 + 0.508i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (0.749 - 0.662i)T \)
53 \( 1 + (0.897 + 0.441i)T \)
59 \( 1 + (0.00951 - 0.999i)T \)
61 \( 1 + (0.398 + 0.917i)T \)
67 \( 1 + (-0.995 - 0.0950i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (0.466 - 0.884i)T \)
79 \( 1 + (-0.879 + 0.475i)T \)
83 \( 1 + (-0.640 - 0.768i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.997 + 0.0760i)T \)
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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

−21.20311257490804767952464139022, −20.47876950550380419400309858889, −19.81773163257976889003056394924, −18.714614140808368490226500209367, −17.9766166216996662643551294620, −17.60476150812062132114671069965, −16.38997520813015919858179146465, −15.36307146431818755089580731999, −14.4905735940630563084559770371, −13.93174649228621223764952247221, −13.282934898263141309050409138090, −12.496567743334324718298787281953, −11.450351554402981802668792219193, −10.76158514823537143832602535194, −10.275540242039540317769165774022, −9.07093548591686838357961002481, −8.585372806729400091681284693472, −7.08158392855537249729592869346, −6.19435183095095246041825115113, −5.3259915190980285375934014616, −4.66290037884777143271989566373, −3.58876211874588243012658799055, −2.60249426012692539341097165239, −1.58281920578784546183321528985, −1.02313334838750087173812852999, 0.95708885551511604670943365892, 2.18115424456221594265610795101, 3.12797101217179526429043669604, 4.3715833366273598086184785382, 5.13989764305625722476619166651, 5.794821471697598902063545364218, 6.66417905395666454449815409945, 7.54094124810474918344431419126, 8.57395463115092169660140039906, 9.05857398613873060288795164270, 10.16843908198103416489326708923, 11.3948226675038694584234661510, 11.85577555375434391918264222748, 13.28180696814807568717555467623, 13.49060890023097885070845319759, 14.254737393083498664671596895560, 15.2037700119796516322919783122, 15.73411948891459974709046663923, 16.84641537788627162047991806933, 17.36123525244064217142809369749, 18.193995825314573075372429559435, 18.57709031552278327659241422974, 20.356038064477482208944630687134, 20.74498328491410595443596137407, 21.52396678257669592870907878416

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