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Properties

Label 1-133-133.83-r1-0-0
Degree $1$
Conductor $133$
Sign $0.305 - 0.952i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + 8-s + (−0.5 − 0.866i)9-s + (0.5 + 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.305 - 0.952i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.305 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329249534 - 0.9692271098i\)
\(L(\frac12)\) \(\approx\) \(1.329249534 - 0.9692271098i\)
\(L(1)\) \(\approx\) \(1.065051635 - 0.2301865437i\)
\(L(1)\) \(\approx\) \(1.065051635 - 0.2301865437i\)

Euler product

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

−28.33327275277416569124719158969, −27.55427767302427694951005350374, −26.77905184107202827695739682854, −25.69610269957192619173102252859, −25.30844176272365758729055426836, −23.12647023983958620221793957448, −21.972226784848010646198356429076, −21.71140269753434983645388353552, −20.40377261451916304206913649535, −19.66665284131121556503826735785, −18.57729747343348117251698827680, −17.50038428240896364205073100750, −16.52437628874014240985319894383, −15.05817963789405006505086850599, −14.14216908401251116492844104883, −13.03272212244248477673973711749, −11.45463986023975682014093797761, −10.57042647820377551387644397096, −9.762419470966670834232007210739, −8.768324460469888012177099581355, −7.52534585646528414900724009644, −5.710841730203433064096579313356, −3.90794084810442651853726912698, −3.15249739262704417420450908932, −1.710654196008211803867083244911, 0.77303868854585844965380616323, 1.90387985658066828389503582331, 4.215072475319959725669115828899, 5.800027830096896743436686448397, 6.70699674987566614897807685359, 7.93229029240371803477048045257, 8.990750769898743501869680725005, 9.5827627923429056092062173659, 11.5671074377014357590019601936, 12.87250024068865215900193741553, 13.93394613759313923744449493210, 14.54450147945611326704416420161, 16.15976646640816874153495707497, 16.94552159946511993868638316848, 17.97487735421931465989770200735, 18.86615532946283567473629616303, 19.88097151545249967658264517768, 20.83441658562819776779532439586, 22.46899821791544170212944789095, 23.68875735952803843651342560252, 24.40266712716337360064161825750, 25.160374631759324407497366622889, 25.84572838654910777399172782765, 27.021333916898197420578135411574, 28.19299238482614882306109467980

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