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Properties

Label 2-176-11.9-c1-0-3
Degree $2$
Conductor $176$
Sign $-0.273 + 0.961i$
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  + (−0.743 + 0.540i)3-s + (−1.24 − 3.82i)5-s + (−3.01 − 2.18i)7-s + (−0.665 + 2.04i)9-s + (1.97 − 2.66i)11-s + (0.324 − 0.998i)13-s + (2.99 + 2.17i)15-s + (−0.291 − 0.898i)17-s + (1.92 − 1.40i)19-s + 3.42·21-s + 1.73·23-s + (−9.05 + 6.58i)25-s + (−1.46 − 4.50i)27-s + (2.22 + 1.61i)29-s + (−1.47 + 4.55i)31-s + ⋯
L(s)  = 1  + (−0.429 + 0.311i)3-s + (−0.556 − 1.71i)5-s + (−1.13 − 0.827i)7-s + (−0.221 + 0.683i)9-s + (0.594 − 0.804i)11-s + (0.0899 − 0.276i)13-s + (0.772 + 0.561i)15-s + (−0.0708 − 0.217i)17-s + (0.442 − 0.321i)19-s + 0.746·21-s + 0.362·23-s + (−1.81 + 1.31i)25-s + (−0.281 − 0.867i)27-s + (0.412 + 0.300i)29-s + (−0.265 + 0.817i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.273 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.273 + 0.961i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ -0.273 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.417239 - 0.552243i\)
\(L(\frac12)\) \(\approx\) \(0.417239 - 0.552243i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.97 + 2.66i)T \)
good3 \( 1 + (0.743 - 0.540i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.24 + 3.82i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.01 + 2.18i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.324 + 0.998i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.291 + 0.898i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.92 + 1.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + (-2.22 - 1.61i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.47 - 4.55i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.71 + 1.24i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.38 + 5.36i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 0.431T + 43T^{2} \)
47 \( 1 + (-5.11 + 3.71i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.0976 + 0.300i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.54 - 4.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.21 + 12.9i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 5.68T + 67T^{2} \)
71 \( 1 + (-3.97 - 12.2i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.84 - 2.06i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.73 + 8.40i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.53 + 13.9i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 2.43T + 89T^{2} \)
97 \( 1 + (-0.457 + 1.40i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.46841480913431735065278230464, −11.48514392381372088551512748830, −10.46907654960237223920439689651, −9.293048451175214255501297095316, −8.502499954918004623434601620660, −7.26184435705957710173773610563, −5.77028632117067388605651632994, −4.73325105820893329435214122761, −3.59167819931955364573235596547, −0.67812747072290170690573639269, 2.71970576039742230833987591170, 3.81931505250471424474722666511, 6.04229052329501929335123120022, 6.58770770870731449405398288657, 7.49317031383483779192281630653, 9.176055859669055409328542619834, 10.01997635704176197463276690571, 11.23265190997190256127775143216, 11.94117014176327566446386236522, 12.69815887555748676302491784615

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