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Properties

Label 2-176-44.43-c1-0-3
Degree $2$
Conductor $176$
Sign $0.866 + 0.5i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.792i·3-s + 1.37·5-s + 2.37·9-s − 3.31i·11-s − 1.08i·15-s + 6.13i·23-s − 3.11·25-s − 4.25i·27-s + 9.30i·31-s − 2.62·33-s − 12.1·37-s + 3.25·45-s + 6.63i·47-s − 7·49-s + 6·53-s + ⋯
L(s)  = 1  − 0.457i·3-s + 0.613·5-s + 0.790·9-s − 1.00i·11-s − 0.280i·15-s + 1.27i·23-s − 0.623·25-s − 0.819i·27-s + 1.67i·31-s − 0.457·33-s − 1.99·37-s + 0.485·45-s + 0.967i·47-s − 49-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.866 + 0.5i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ 0.866 + 0.5i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24970 - 0.334856i\)
\(L(\frac12)\) \(\approx\) \(1.24970 - 0.334856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + 3.31iT \)
good3 \( 1 + 0.792iT - 3T^{2} \)
5 \( 1 - 1.37T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6.13iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 9.30iT - 31T^{2} \)
37 \( 1 + 12.1T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 6.63iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 14.6iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 16.2iT - 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.8T + 89T^{2} \)
97 \( 1 + 0.116T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.73564373406639655592685129563, −11.72245059291821505331365226180, −10.59388607790233253276999367049, −9.656254189946741682021030247356, −8.545077866047027119965392549875, −7.36446723234155150585542721975, −6.31941455450258019569995334222, −5.18262010759134743285302351250, −3.46856806928306764618609269897, −1.62860371650146495490960633779, 2.07051252179643636702352886208, 4.00216064520592543125067253567, 5.09749591446562663117408713672, 6.47809816408198717520290975700, 7.55691026273009723830964780581, 8.971505389799373493866520546261, 9.938977076015503316310252348926, 10.47291822819117213724835335424, 11.88345368670506347705151222114, 12.82018501358825780657637554935

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