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Properties

Label 2-176-11.4-c3-0-11
Degree $2$
Conductor $176$
Sign $0.297 + 0.954i$
Analytic cond. $10.3843$
Root an. cond. $3.22247$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.320 + 0.986i)3-s + (−7.72 + 5.61i)5-s + (−5.56 − 17.1i)7-s + (20.9 + 15.2i)9-s + (−10.0 − 35.0i)11-s + (−13.3 − 9.68i)13-s + (−3.05 − 9.41i)15-s + (64.6 − 46.9i)17-s + (44.2 − 136. i)19-s + 18.6·21-s + 45.6·23-s + (−10.4 + 32.2i)25-s + (−44.4 + 32.2i)27-s + (−45.4 − 140. i)29-s + (−12.6 − 9.17i)31-s + ⋯
L(s)  = 1  + (−0.0616 + 0.189i)3-s + (−0.690 + 0.501i)5-s + (−0.300 − 0.924i)7-s + (0.776 + 0.564i)9-s + (−0.274 − 0.961i)11-s + (−0.284 − 0.206i)13-s + (−0.0526 − 0.162i)15-s + (0.921 − 0.669i)17-s + (0.534 − 1.64i)19-s + 0.194·21-s + 0.413·23-s + (−0.0837 + 0.257i)25-s + (−0.316 + 0.229i)27-s + (−0.291 − 0.896i)29-s + (−0.0731 − 0.0531i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.297 + 0.954i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ 0.297 + 0.954i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.956501 - 0.703703i\)
\(L(\frac12)\) \(\approx\) \(0.956501 - 0.703703i\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.0 + 35.0i)T \)
good3 \( 1 + (0.320 - 0.986i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (7.72 - 5.61i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (5.56 + 17.1i)T + (-277. + 201. i)T^{2} \)
13 \( 1 + (13.3 + 9.68i)T + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (-64.6 + 46.9i)T + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-44.2 + 136. i)T + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 - 45.6T + 1.21e4T^{2} \)
29 \( 1 + (45.4 + 140. i)T + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (12.6 + 9.17i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (89.0 + 273. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-92.6 + 285. i)T + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 125.T + 7.95e4T^{2} \)
47 \( 1 + (-9.79 + 30.1i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-421. - 306. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (-70.9 - 218. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (600. - 436. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 505.T + 3.00e5T^{2} \)
71 \( 1 + (689. - 500. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-74.0 - 227. i)T + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (592. + 430. i)T + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (-476. + 346. i)T + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 663.T + 7.04e5T^{2} \)
97 \( 1 + (-1.10e3 - 801. i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−11.86462501512586601281868852023, −10.94039168343548931706362078672, −10.27893854919957388700818579951, −9.086419293204217774496172744135, −7.45951624810101428532938071668, −7.26377350636258691267845811113, −5.46909326626643159797406287324, −4.14729156292572707730024778336, −2.98346245487985963591247632821, −0.58115764891434209129361967860, 1.54683328105837874976337028845, 3.47223625423634239405622390863, 4.78391568656782545365153584315, 6.07286963843566946165143025940, 7.36149947522045607615683055992, 8.275238483230031548735290912970, 9.497061530359439463976623626205, 10.25932877855785992076880617525, 12.06479575147918671126056843609, 12.17088200343284247576098219980

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