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Properties

Label 2-176-11.6-c2-0-5
Degree $2$
Conductor $176$
Sign $0.805 - 0.592i$
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  + (4.08 + 2.97i)3-s + (1.46 − 4.50i)5-s + (3.91 + 5.39i)7-s + (5.10 + 15.7i)9-s + (−6.32 − 9.00i)11-s + (14.3 − 4.65i)13-s + (19.3 − 14.0i)15-s + (−16.9 − 5.52i)17-s + (−19.0 + 26.1i)19-s + 33.6i·21-s + 3.58·23-s + (2.04 + 1.48i)25-s + (−11.7 + 36.2i)27-s + (−7.50 − 10.3i)29-s + (−6.76 − 20.8i)31-s + ⋯
L(s)  = 1  + (1.36 + 0.990i)3-s + (0.293 − 0.901i)5-s + (0.559 + 0.770i)7-s + (0.567 + 1.74i)9-s + (−0.574 − 0.818i)11-s + (1.10 − 0.357i)13-s + (1.29 − 0.938i)15-s + (−0.999 − 0.324i)17-s + (−1.00 + 1.37i)19-s + 1.60i·21-s + 0.156·23-s + (0.0816 + 0.0592i)25-s + (−0.435 + 1.34i)27-s + (−0.258 − 0.356i)29-s + (−0.218 − 0.671i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.805 - 0.592i$
Analytic conductor: \(4.79565\)
Root analytic conductor: \(2.18989\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1),\ 0.805 - 0.592i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.27100 + 0.745233i\)
\(L(\frac12)\) \(\approx\) \(2.27100 + 0.745233i\)
\(L(2)\) 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 \( 1 + (6.32 + 9.00i)T \)
good3 \( 1 + (-4.08 - 2.97i)T + (2.78 + 8.55i)T^{2} \)
5 \( 1 + (-1.46 + 4.50i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (-3.91 - 5.39i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (-14.3 + 4.65i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (16.9 + 5.52i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (19.0 - 26.1i)T + (-111. - 343. i)T^{2} \)
23 \( 1 - 3.58T + 529T^{2} \)
29 \( 1 + (7.50 + 10.3i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (6.76 + 20.8i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (-22.4 + 16.2i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (29.1 - 40.0i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 + 16.8iT - 1.84e3T^{2} \)
47 \( 1 + (63.0 + 45.8i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-3.06 - 9.41i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-29.6 + 21.5i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (26.9 + 8.74i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 30.7T + 4.48e3T^{2} \)
71 \( 1 + (-12.6 + 39.0i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (36.3 + 49.9i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (118. - 38.4i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-2.94 - 0.956i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 - 60.4T + 7.92e3T^{2} \)
97 \( 1 + (-14.8 - 45.6i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−12.98291743569626937401008683198, −11.40418653338596517910353302985, −10.40026893286873565120121287004, −9.278940532613401237360451695288, −8.475948375288388748898390020549, −8.189149315397245042566738156639, −5.88538369297446988395421980117, −4.76169769578713731810963550185, −3.54615376212845005359259548791, −2.07956297410199193584629502119, 1.74367606147749083446988587576, 2.86643998241295459988294250665, 4.35495673973311382728311799356, 6.61991885246444771806974354288, 7.10040815286337837435642394803, 8.230332821372581660278299945315, 9.052329134665939077841640872126, 10.46132106211160510203878330661, 11.23564408022543122798404659632, 12.90987031925461686090346181936

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