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Properties

Label 1-133-133.109-r1-0-0
Degree $1$
Conductor $133$
Sign $0.993 - 0.117i$
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  + (−0.173 − 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.939 + 0.342i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.939 − 0.342i)5-s + (−0.939 + 0.342i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.173 + 0.984i)10-s + (−0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + (0.939 − 0.342i)13-s + (−0.173 + 0.984i)15-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.993 - 0.117i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.993 - 0.117i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5003387464 + 0.02942905175i\)
\(L(\frac12)\) \(\approx\) \(0.5003387464 + 0.02942905175i\)
\(L(1)\) \(\approx\) \(0.4936292289 - 0.3683732114i\)
\(L(1)\) \(\approx\) \(0.4936292289 - 0.3683732114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.766 - 0.642i)T \)
31 \( 1 - T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (-0.766 + 0.642i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.766 + 0.642i)T \)
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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

−27.99862309474392101880272254116, −27.172574546580620784414541027287, −26.45109537827143581480324297077, −25.78066241783274397999406136580, −24.219499940707891221730983720, −23.46039770067246517911835176939, −22.55186530153216143352617661849, −21.67272141279305863385352182183, −20.32667758061850800787521047840, −19.05861429935183729820315214307, −18.153087461758786213301157117653, −16.73900168079293295820646356494, −16.06116118568160284698188558910, −15.281864591934868383756321726771, −14.39501053027624780873889291147, −13.08429275658814766655816535042, −11.26632992041949095775107869748, −10.58423154313320192865471970410, −8.9915399903450021616254815147, −8.32657307061190776632001984478, −6.85420988829905768287828860357, −5.66532805696909827577184165487, −4.38862802201736263752239448161, −3.41596374164072654880342395127, −0.267051417697106983176985368130, 1.0797812848553048723676537409, 2.51766238355344573741833859822, 3.992664013680190017302944862327, 5.36135341489824125634882606488, 7.236379104720406186810604838302, 8.14440881557700525060436755116, 9.237231083887792879414322403754, 10.97997507188248086102739952587, 11.55689249853038978695172973619, 12.85529603334960354692885741057, 13.17815008799091868388996581205, 14.81265508779835013990904886523, 16.25981960948772952201959774077, 17.63831907956873593641345948377, 18.29769747598190393284064995504, 19.37890857371400177285204036267, 20.08687085522744149469592533991, 20.98088882038132840769714957549, 22.68703018093370035601459964895, 23.10992081849907684559908988381, 24.1267403526926957204739982353, 25.43643471158185140924433496850, 26.52700192908394091256389553066, 27.79336861365713775299852140742, 28.37300139312716746386292214889

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