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Properties

Label 1-133-133.26-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.182 + 0.983i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s − 5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s − 5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 + 0.866i)9-s − 10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + 16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.182 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.182 + 0.983i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.182 + 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.025222640 + 2.436888020i\)
\(L(\frac12)\) \(\approx\) \(2.025222640 + 2.436888020i\)
\(L(1)\) \(\approx\) \(1.755779632 + 0.8809259784i\)
\(L(1)\) \(\approx\) \(1.755779632 + 0.8809259784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 - T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−28.36584132235000581101829867568, −26.96045291940601827950106560192, −25.81280530749516536760130025862, −24.823965738481802341096858992818, −23.98028921072752343061361150911, −23.278900656655677730137808374618, −22.37618240593037066589086037619, −20.82836889129026380973098009445, −20.18227396163251011794945352092, −19.16053386064570524647804934715, −18.24818346026996738919193801494, −16.46983570705747540935672168748, −15.57420665897525106736365749510, −14.51427792385760767135835265371, −13.560649952942703419201697243220, −12.58415212956268706230520756187, −11.73663774586663299910414759300, −10.62935455457671467646635793109, −8.47267051967314946754655651280, −7.67269768526316031222489936721, −6.55754024594037163026566516460, −5.23710814958353449850096957195, −3.57899956341886128417086661456, −2.79282016114330670087827959481, −0.87235189754082043037828417994, 2.17510581340327114939299470125, 3.70882077894804966938356523042, 4.26800148330150504520488266897, 5.59578862694211051061456622245, 7.271125135826073078975804370521, 8.2659299374454678169122558602, 9.88539393985363005103146620504, 11.03491371015340329742195202649, 11.96858805191850826916156311690, 13.23331402672632653816124280738, 14.41226550162327748903959504542, 15.29193960571684278799870898155, 15.90010031999171657818371569753, 16.96407153925477786257085707641, 19.00252024389165954676006964752, 19.86144352296259193398639608452, 20.79089233088447950547060591904, 21.52988186782651573111025723983, 22.75197596165045308459048245593, 23.41181631946507187999007364495, 24.49167923394385528687692491082, 25.85529433443188434953475815752, 26.333020917031554597627628926762, 27.91565900605187553137707281803, 28.38043568241397505470795441401

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