Properties

Label 1-176-176.43-r0-0-0
Degree $1$
Conductor $176$
Sign $-0.923 + 0.382i$
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  + i·3-s + i·5-s − 7-s − 9-s + i·13-s − 15-s − 17-s i·19-s i·21-s + 23-s − 25-s i·27-s + i·29-s − 31-s i·35-s + ⋯
L(s)  = 1  + i·3-s + i·5-s − 7-s − 9-s + i·13-s − 15-s − 17-s i·19-s i·21-s + 23-s − 25-s i·27-s + i·29-s − 31-s i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(0.817340\)
Root analytic conductor: \(0.817340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 176,\ (0:\ ),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1431418629 + 0.7196227406i\)
\(L(\frac12)\) \(\approx\) \(0.1431418629 + 0.7196227406i\)
\(L(1)\) \(\approx\) \(0.6578181277 + 0.4918640942i\)
\(L(1)\) \(\approx\) \(0.6578181277 + 0.4918640942i\)

Euler product

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

−27.033416358563142311037433187, −25.70777502045407098723339791935, −24.97899970606589220466827646299, −24.320533052414401496208134452, −23.12826782443184075455924284496, −22.54758989493292054216768761399, −20.994818882448632014135988702106, −19.97320839928255124181585132430, −19.39256340352040011855005778019, −18.21679200873334594183503049656, −17.22950856037729545345096557396, −16.35919806844901404137221090168, −15.21855687570203449668433601352, −13.73360323619356118730213215691, −12.85340702244536451685519752918, −12.445966381247016981647956250534, −11.04962582748981728983233728383, −9.56307898319615970377219937818, −8.56582060639574182019149942480, −7.52342102865042222734616476843, −6.28059288318391360449107392771, −5.32862987009681023260377676926, −3.62180028563855415485403038752, −2.154141173145501248505822649431, −0.56821652205539802518194921902, 2.57124949016847090509462538891, 3.52014546652345124015947751650, 4.73659231689968938102561235178, 6.259604646955947293719644353880, 7.06714280608486547585834792060, 8.91496381345808561667690506976, 9.615557326454500924097151029920, 10.79423905366983467046342172065, 11.429869381397273587368089421891, 13.05623527804352003708972520544, 14.17895381109361488989417722794, 15.132917796625731965455937287315, 15.94804883952228286514254677726, 16.89536163907948010956470606817, 18.09363824964794946248529007250, 19.26956478543737781769499644831, 19.98504766101790087843434947939, 21.387566660151841476596415184132, 22.062130020389173188764882868058, 22.74655043801682086439119297360, 23.77846920937473303542463342640, 25.35801912369816238167615672043, 26.20852338451153515359900740553, 26.59483834114874586121545805105, 27.73731151819277102599674825664

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