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Properties

Label 1-176-176.83-r0-0-0
Degree $1$
Conductor $176$
Sign $0.745 + 0.666i$
Analytic cond. $0.817340$
Root an. cond. $0.817340$
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  + (0.951 − 0.309i)3-s + (−0.587 + 0.809i)5-s + (−0.309 + 0.951i)7-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.951 + 0.309i)19-s − i·21-s + 23-s + (−0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)35-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)3-s + (−0.587 + 0.809i)5-s + (−0.309 + 0.951i)7-s + (0.809 − 0.587i)9-s + (0.587 + 0.809i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (−0.951 + 0.309i)19-s − i·21-s + 23-s + (−0.309 − 0.951i)25-s + (0.587 − 0.809i)27-s + (−0.951 − 0.309i)29-s + (0.809 − 0.587i)31-s + (−0.587 − 0.809i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.745 + 0.666i$
Analytic conductor: \(0.817340\)
Root analytic conductor: \(0.817340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 176,\ (0:\ ),\ 0.745 + 0.666i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.296134289 + 0.4944231703i\)
\(L(\frac12)\) \(\approx\) \(1.296134289 + 0.4944231703i\)
\(L(1)\) \(\approx\) \(1.255097079 + 0.2246800765i\)
\(L(1)\) \(\approx\) \(1.255097079 + 0.2246800765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.951 - 0.309i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
13 \( 1 + (0.587 + 0.809i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.951 - 0.309i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.587 + 0.809i)T \)
59 \( 1 + (0.951 + 0.309i)T \)
61 \( 1 + (-0.587 + 0.809i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.587 - 0.809i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−27.408596132665264504014434701109, −26.31323123022234766624764535859, −25.46155904454813855020704839015, −24.55045656055965868541117553586, −23.48344385784408325775023711109, −22.66011159944509160622324149068, −20.99966320602715403809769816425, −20.63669283978393011407491704229, −19.63429536687948896826423428402, −18.955418605587957597656629627761, −17.33781434627564830363028291018, −16.32138579857711808442367720288, −15.564520890700112298497059584674, −14.460600913219159769636176345687, −13.31008727799274801301543898105, −12.690680031935038760465231585981, −11.07181172849811193683793414766, −10.02677004079225044646528428582, −8.89550298713085052084517638982, −8.007956316618966076836581864703, −7.0319598361680353418064854114, −5.13012466528775816024899174565, −4.018737051271669130563780647696, −3.09569856257089845998859122432, −1.15373018883373719059496813108, 1.93389154919065328826768431186, 3.09329205236338307558440427917, 4.07193825302120327902181633398, 6.04267110428003334047249572411, 7.046828778236242166791371511922, 8.211826370394965050194265489740, 9.05837677782602124682232030376, 10.30948215620002322500328614271, 11.63483629369156984878256907700, 12.60098406204527700013224260618, 13.73946474511165209609192290982, 14.92112467557899412511930428520, 15.26688902919600647930796686120, 16.60702977432924351969547776756, 18.23274791889557168229114115927, 19.033631985952707213795370606960, 19.31770196745018829967457296015, 20.86951913880386028040906442288, 21.5561085589345904510264021160, 22.82614588792555706124074198673, 23.7165682318561603963636074954, 24.81766088184709815208323637798, 25.770931212062978833720678770974, 26.28719538511719658636505661457, 27.42893437772540222395672462484

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