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Properties

Label 2-176-44.7-c1-0-0
Degree $2$
Conductor $176$
Sign $0.180 - 0.983i$
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  + (−2.94 − 0.957i)3-s + (−0.540 + 0.392i)5-s + (0.818 + 2.52i)7-s + (5.34 + 3.88i)9-s + (1.92 + 2.69i)11-s + (−3.44 + 4.74i)13-s + (1.96 − 0.639i)15-s + (−1.86 − 2.56i)17-s + (0.373 − 1.14i)19-s − 8.21i·21-s + 1.68i·23-s + (−1.40 + 4.33i)25-s + (−6.56 − 9.03i)27-s + (5.40 − 1.75i)29-s + (−3.31 + 4.55i)31-s + ⋯
L(s)  = 1  + (−1.70 − 0.552i)3-s + (−0.241 + 0.175i)5-s + (0.309 + 0.952i)7-s + (1.78 + 1.29i)9-s + (0.581 + 0.813i)11-s + (−0.955 + 1.31i)13-s + (0.508 − 0.165i)15-s + (−0.452 − 0.622i)17-s + (0.0856 − 0.263i)19-s − 1.79i·21-s + 0.352i·23-s + (−0.281 + 0.866i)25-s + (−1.26 − 1.73i)27-s + (1.00 − 0.325i)29-s + (−0.594 + 0.818i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.180 - 0.983i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ 0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410876 + 0.342238i\)
\(L(\frac12)\) \(\approx\) \(0.410876 + 0.342238i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.92 - 2.69i)T \)
good3 \( 1 + (2.94 + 0.957i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.540 - 0.392i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.818 - 2.52i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.44 - 4.74i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.86 + 2.56i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.373 + 1.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.68iT - 23T^{2} \)
29 \( 1 + (-5.40 + 1.75i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.31 - 4.55i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.283 - 0.873i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (9.56 + 3.10i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 5.75T + 43T^{2} \)
47 \( 1 + (-8.10 - 2.63i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.20 - 1.60i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-6.96 + 2.26i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.13 - 8.45i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 2.26iT - 67T^{2} \)
71 \( 1 + (5.66 + 7.79i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.14 + 0.697i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (0.845 + 0.614i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.45 - 6.14i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 7.11T + 89T^{2} \)
97 \( 1 + (0.190 + 0.138i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.42448443983296861053674440227, −11.81458608726242537273108263393, −11.46012306429915618221130645993, −10.11844134935199713950591557714, −8.999374251279131152284678781590, −7.22791872277798523757533719330, −6.74119813864603291067129887208, −5.40292250157865675466649746040, −4.56885091488465913327827346910, −1.91584055308820619197847304009, 0.61679077220541269669104542034, 3.87257882158357643319531360015, 4.89956547577711535517398365075, 5.94855532979253813112117294956, 7.01731371006051906021130871934, 8.339981192723207501212904356536, 10.05337875257672913188221039878, 10.50058636773451497219333526522, 11.45451603840489144944757480774, 12.21113335140560177539815325555

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