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Properties

Label 2-176-11.9-c1-0-2
Degree $2$
Conductor $176$
Sign $0.998 - 0.0540i$
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  + (1.55 − 1.12i)3-s + (1.05 + 3.23i)5-s + (0.703 + 0.510i)7-s + (0.211 − 0.649i)9-s + (−3.16 − 0.998i)11-s + (0.866 − 2.66i)13-s + (5.28 + 3.84i)15-s + (−2.25 − 6.93i)17-s + (1.92 − 1.40i)19-s + 1.66·21-s − 7.44·23-s + (−5.34 + 3.88i)25-s + (1.37 + 4.22i)27-s + (5.93 + 4.31i)29-s + (0.816 − 2.51i)31-s + ⋯
L(s)  = 1  + (0.896 − 0.651i)3-s + (0.470 + 1.44i)5-s + (0.265 + 0.193i)7-s + (0.0703 − 0.216i)9-s + (−0.953 − 0.301i)11-s + (0.240 − 0.739i)13-s + (1.36 + 0.992i)15-s + (−0.546 − 1.68i)17-s + (0.442 − 0.321i)19-s + 0.363·21-s − 1.55·23-s + (−1.06 + 0.776i)25-s + (0.264 + 0.813i)27-s + (1.10 + 0.801i)29-s + (0.146 − 0.451i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.998 - 0.0540i$
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: \((2,\ 176,\ (\ :1/2),\ 0.998 - 0.0540i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53781 + 0.0415982i\)
\(L(\frac12)\) \(\approx\) \(1.53781 + 0.0415982i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.16 + 0.998i)T \)
good3 \( 1 + (-1.55 + 1.12i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-1.05 - 3.23i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.703 - 0.510i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.866 + 2.66i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.25 + 6.93i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.92 + 1.40i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.44T + 23T^{2} \)
29 \( 1 + (-5.93 - 4.31i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.816 + 2.51i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.834 + 0.605i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (2.34 - 1.70i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.18T + 43T^{2} \)
47 \( 1 + (3.19 - 2.31i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.19 - 6.76i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.54 - 4.75i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.832 + 2.56i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 13.7T + 67T^{2} \)
71 \( 1 + (-0.596 - 1.83i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.03 + 2.93i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.15 - 12.7i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.0601 - 0.185i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 4.18T + 89T^{2} \)
97 \( 1 + (-1.54 + 4.74i)T + (-78.4 - 57.0i)T^{2} \)
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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

−13.05315944402905315422014331648, −11.63716420780627309193661972249, −10.69005602900050560794142251733, −9.789877045761994181941729613396, −8.408379037558050575646428536633, −7.57239482989204143063288417543, −6.65781483595057906904244684837, −5.29299658121973665689676484242, −3.06307697149488688367051117073, −2.40122090864405302219845961767, 1.93936818190157136970522724158, 3.90499052996478241306911799024, 4.82563570989523183668092620414, 6.17209020383845571873053550065, 8.167007732314557124133852004242, 8.512308732122565098309263123687, 9.659190048505573160932941860727, 10.31660189611471297031111461162, 11.89015930925937246650115578524, 12.83974682082427948928847735517

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