Properties

Label 1-176-176.131-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: $\chi_{176} (131, \cdot )$
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

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

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