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Properties

Label 1-133-133.53-r1-0-0
Degree $1$
Conductor $133$
Sign $0.491 - 0.870i$
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  + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.939 + 0.342i)10-s + 11-s − 12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (−0.766 − 0.642i)3-s + (0.766 − 0.642i)4-s + (0.766 + 0.642i)5-s + (−0.939 − 0.342i)6-s + (0.5 − 0.866i)8-s + (0.173 + 0.984i)9-s + (0.939 + 0.342i)10-s + 11-s − 12-s + (0.939 + 0.342i)13-s + (−0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (0.173 − 0.984i)17-s + (0.5 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.491 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $0.491 - 0.870i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ 0.491 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.680556194 - 1.564193001i\)
\(L(\frac12)\) \(\approx\) \(2.680556194 - 1.564193001i\)
\(L(1)\) \(\approx\) \(1.740599387 - 0.6518204895i\)
\(L(1)\) \(\approx\) \(1.740599387 - 0.6518204895i\)

Euler product

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

−28.5121659146896994378197542880, −27.837140293190714431132099514482, −26.33232188970100330702700725535, −25.4029362301435199195760459028, −24.39181562897699612125327691454, −23.46335567785673836014491140560, −22.465226858641449553207972179, −21.63627844590680124889105802517, −20.93201719130116795944616035073, −19.90776491742703356673319603162, −17.872115125459461101250975965337, −17.03478233981920133408454673788, −16.28802598388977154860493875139, −15.24152156449735412362084011499, −14.10381797059121879397492917513, −12.96609230349209820216171964247, −12.01513599158554229456208371522, −10.934153493420083930657127272819, −9.66105077440045038301691885407, −8.301254621554999420525950259836, −6.36348400178262720287852269962, −5.86585990116098218234914264305, −4.60476096201773618150102627852, −3.59591145845091996824224995540, −1.4821433179957163069770004550, 1.22448995030948684462725051945, 2.43293534715376557998314968812, 4.08136095539723843918101481547, 5.63756420678379513313180038879, 6.33627394481997776191344252294, 7.307976850038778195139771662372, 9.49169017421827391388865454552, 10.80649942356409884654537445944, 11.52240436423409278602239956890, 12.616295692505551152575564854578, 13.764188583226621908607786271176, 14.30489615656680396078615445234, 15.88515979079439463412840264789, 16.96088581515812357777756446502, 18.24002086039122166123996083841, 18.975668744929517167709927530471, 20.30094758818570697222832097339, 21.49185097896901966334714584811, 22.39597511129785078915546217986, 22.91172644942724655957738361215, 24.11637240499628991375749107153, 24.9331413265020042217971082648, 25.85482016210138904899567233620, 27.622775598247763709870100763228, 28.53865718201013409386568438840

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