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Properties

Label 1-33-33.26-r1-0-0
Degree $1$
Conductor $33$
Sign $0.286 + 0.958i$
Analytic cond. $3.54634$
Root an. cond. $3.54634$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + 10-s + (0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)26-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.809 + 0.587i)8-s + 10-s + (0.309 + 0.951i)13-s + (0.809 + 0.587i)14-s + (0.309 − 0.951i)16-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s + (−0.309 − 0.951i)20-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(3.54634\)
Root analytic conductor: \(3.54634\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{33} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 33,\ (1:\ ),\ 0.286 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5261439519 + 0.3919053075i\)
\(L(\frac12)\) \(\approx\) \(0.5261439519 + 0.3919053075i\)
\(L(1)\) \(\approx\) \(0.6757911734 + 0.01606923377i\)
\(L(1)\) \(\approx\) \(0.6757911734 + 0.01606923377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−35.81948766187763837222546933307, −34.82128810199806968801176508700, −33.316028070791259580326967607102, −32.35581150946625242963748888450, −31.509423262127814779872612419393, −29.52896202542071772794494193256, −28.14107181403048900249549179244, −27.17412840167495256736247104830, −25.8010372090391135679199935037, −24.754051968135648567631050240767, −23.52015527731678349839060231769, −22.54783187711255055087900285271, −20.44563548047699495859088356477, −19.28913780760915975143500887992, −17.67594286701284637561003135733, −16.44464231274880934344363843143, −15.619711176443809733458027314309, −13.80545834652201277209210890442, −12.618080628922165256608269575348, −10.27487143468157389613533549142, −8.87025446251755237575641296318, −7.52135083655337149435006247143, −5.846651711501946371697316531893, −4.16962692738494654040016646398, −0.50071745761369093302782530245, 2.419982580182186069868696470257, 3.95673334488930028563352422346, 6.52285903884478324251886531972, 8.49968743776815773146501554292, 9.99032391410647686603566622895, 11.267207605513250484970236411278, 12.55573575193658256040469067363, 14.09624089983057919276798646536, 15.79861083486147757458609670763, 17.58726618088389042027930084516, 18.93115591729571247667344564796, 19.53767943171762349287135813403, 21.41364389096790088521889805526, 22.25334557821457509314652370446, 23.50403908164625079894409898924, 25.75954099212442504176262393819, 26.46894457735918296835459069117, 27.95064702564972100516442296674, 28.958037385133281467682645552700, 30.22708015027742760761402810196, 31.13846793975242555690599343245, 32.37045645828092889692794969340, 34.24702576596132846471141940265, 35.26489091769094574770587349929, 36.41498338478273082726736301461

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