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Properties

Label 2-176-11.2-c2-0-0
Degree $2$
Conductor $176$
Sign $-0.968 + 0.247i$
Analytic cond. $4.79565$
Root an. cond. $2.18989$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.27 + 1.65i)3-s + (−0.0380 − 0.117i)5-s + (−1.51 + 2.09i)7-s + (−0.328 + 1.01i)9-s + (−2.96 − 10.5i)11-s + (−17.5 − 5.69i)13-s + (0.280 + 0.203i)15-s + (−22.4 + 7.28i)17-s + (−1.42 − 1.96i)19-s − 7.28i·21-s − 21.8·23-s + (20.2 − 14.6i)25-s + (−8.76 − 26.9i)27-s + (4.65 − 6.40i)29-s + (−14.2 + 43.9i)31-s + ⋯
L(s)  = 1  + (−0.759 + 0.551i)3-s + (−0.00760 − 0.0234i)5-s + (−0.217 + 0.298i)7-s + (−0.0365 + 0.112i)9-s + (−0.269 − 0.963i)11-s + (−1.34 − 0.437i)13-s + (0.0187 + 0.0135i)15-s + (−1.31 + 0.428i)17-s + (−0.0750 − 0.103i)19-s − 0.346i·21-s − 0.951·23-s + (0.808 − 0.587i)25-s + (−0.324 − 0.998i)27-s + (0.160 − 0.220i)29-s + (−0.460 + 1.41i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.968 + 0.247i$
Analytic conductor: \(4.79565\)
Root analytic conductor: \(2.18989\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1),\ -0.968 + 0.247i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0145270 - 0.115392i\)
\(L(\frac12)\) \(\approx\) \(0.0145270 - 0.115392i\)
\(L(2)\) 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 \( 1 + (2.96 + 10.5i)T \)
good3 \( 1 + (2.27 - 1.65i)T + (2.78 - 8.55i)T^{2} \)
5 \( 1 + (0.0380 + 0.117i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (1.51 - 2.09i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (17.5 + 5.69i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (22.4 - 7.28i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (1.42 + 1.96i)T + (-111. + 343. i)T^{2} \)
23 \( 1 + 21.8T + 529T^{2} \)
29 \( 1 + (-4.65 + 6.40i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (14.2 - 43.9i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (26.0 + 18.9i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-31.1 - 42.9i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 66.1iT - 1.84e3T^{2} \)
47 \( 1 + (-34.5 + 25.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (-13.1 + 40.4i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-47.2 - 34.3i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (61.3 - 19.9i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 20.9T + 4.48e3T^{2} \)
71 \( 1 + (19.1 + 58.9i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-37.4 + 51.5i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (36.1 + 11.7i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (49.8 - 16.1i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 27.8T + 7.92e3T^{2} \)
97 \( 1 + (51.8 - 159. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−12.84045522827234879979508282532, −11.93654357427295490578914828103, −10.88374327361576691212221965475, −10.29834262159877374342078112214, −9.048850056951644981132470535481, −7.955252917675195040368013966137, −6.49021408360753481867867352735, −5.44814997338899163503565104775, −4.45037086650128273272090827395, −2.65678069350323393705380876669, 0.07094348028464617928541128619, 2.21619261918648421335025153461, 4.26435860987783377306192795608, 5.49392782421241902053072114523, 6.86536161530314788885261815462, 7.32242457061733898701054878948, 9.014153970961905300347628926458, 9.986034265101395627922030536393, 11.09888264340302910727219532511, 12.09574409634167725184477169620

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