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Properties

Label 1-33e2-1089.212-r1-0-0
Degree $1$
Conductor $1089$
Sign $-0.0524 + 0.998i$
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  + (0.398 − 0.917i)2-s + (−0.683 − 0.730i)4-s + (0.432 − 0.901i)5-s + (0.272 + 0.962i)7-s + (−0.941 + 0.336i)8-s + (−0.654 − 0.755i)10-s + (0.483 − 0.875i)13-s + (0.991 + 0.132i)14-s + (−0.0665 + 0.997i)16-s + (−0.774 − 0.633i)17-s + (−0.998 + 0.0570i)19-s + (−0.953 + 0.299i)20-s + (0.786 − 0.618i)23-s + (−0.625 − 0.780i)25-s + (−0.610 − 0.791i)26-s + ⋯
L(s)  = 1  + (0.398 − 0.917i)2-s + (−0.683 − 0.730i)4-s + (0.432 − 0.901i)5-s + (0.272 + 0.962i)7-s + (−0.941 + 0.336i)8-s + (−0.654 − 0.755i)10-s + (0.483 − 0.875i)13-s + (0.991 + 0.132i)14-s + (−0.0665 + 0.997i)16-s + (−0.774 − 0.633i)17-s + (−0.998 + 0.0570i)19-s + (−0.953 + 0.299i)20-s + (0.786 − 0.618i)23-s + (−0.625 − 0.780i)25-s + (−0.610 − 0.791i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0524 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1089\)    =    \(3^{2} \cdot 11^{2}\)
Sign: $-0.0524 + 0.998i$
Analytic conductor: \(117.029\)
Root analytic conductor: \(117.029\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1089} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1089,\ (1:\ ),\ -0.0524 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3698634132 - 0.3898110143i\)
\(L(\frac12)\) \(\approx\) \(-0.3698634132 - 0.3898110143i\)
\(L(1)\) \(\approx\) \(0.8329947917 - 0.6964019267i\)
\(L(1)\) \(\approx\) \(0.8329947917 - 0.6964019267i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.398 - 0.917i)T \)
5 \( 1 + (0.432 - 0.901i)T \)
7 \( 1 + (0.272 + 0.962i)T \)
13 \( 1 + (0.483 - 0.875i)T \)
17 \( 1 + (-0.774 - 0.633i)T \)
19 \( 1 + (-0.998 + 0.0570i)T \)
23 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (-0.988 - 0.151i)T \)
31 \( 1 + (0.797 + 0.603i)T \)
37 \( 1 + (0.0855 + 0.996i)T \)
41 \( 1 + (-0.00951 - 0.999i)T \)
43 \( 1 + (0.723 - 0.690i)T \)
47 \( 1 + (0.948 + 0.318i)T \)
53 \( 1 + (-0.897 - 0.441i)T \)
59 \( 1 + (-0.861 - 0.508i)T \)
61 \( 1 + (-0.595 + 0.803i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.254 - 0.967i)T \)
73 \( 1 + (-0.466 + 0.884i)T \)
79 \( 1 + (-0.851 - 0.524i)T \)
83 \( 1 + (-0.345 + 0.938i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.432 - 0.901i)T \)
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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

−21.70883340090422249637248906885, −21.45074443220784365743916182217, −20.46170807563733007131482735856, −19.24433642439350488815755542216, −18.63139249898805396845287539318, −17.578384748901697772011792649628, −17.23525453053036088539089123877, −16.42593837443806613353052584844, −15.37928355328232215733574221152, −14.78230339581117146728115939634, −14.0283372763138067582824757757, −13.42809122298732636328873183087, −12.75016394534024740466394196009, −11.30630682610426335837127480367, −10.881111404858583278013264932631, −9.70987693539665211532520261156, −8.91407543346238351418764476856, −7.87228935546362420898273994027, −7.09072672349140059442697779723, −6.46785947467597436276536226251, −5.73988475502404021186519149989, −4.40652201405868076436719529599, −3.95250944286423688037608552906, −2.80026275116415463387526837276, −1.55644051515657630219324021068, 0.09407728438889131472035056248, 1.17581476384833370581759167931, 2.163282554636998169776356962540, 2.89741119677259003377460790992, 4.19156987594767801983274115428, 4.977580494083363094703742288313, 5.635579504481711259545347477271, 6.47536302524851772923697982380, 8.16202371541833904680203428135, 8.83761113702449414606998864046, 9.36498543553422460899836589419, 10.4710282756307082414041285886, 11.15118094340524645038217650953, 12.15811551527672902437691469437, 12.67625024473043223244543710120, 13.35984579256468894870254365946, 14.15896665434046718383588637667, 15.24077282937863125926864973994, 15.67017087877890559607582367250, 17.02859565080336065561294989618, 17.67568101948602895277898385852, 18.50520177420611549096145097876, 19.14201873909502163514251452647, 20.18517288048737145671177184480, 20.74148300456791496216874246627

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