Properties

Label 2-176-11.5-c3-0-6
Degree $2$
Conductor $176$
Sign $0.0521 - 0.998i$
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  + (6.12 + 4.45i)3-s + (1.67 − 5.14i)5-s + (−17.9 + 13.0i)7-s + (9.39 + 28.9i)9-s + (11.0 + 34.7i)11-s + (23.7 + 73.0i)13-s + (33.1 − 24.0i)15-s + (18.3 − 56.3i)17-s + (77.0 + 55.9i)19-s − 167.·21-s − 142.·23-s + (77.4 + 56.2i)25-s + (−7.94 + 24.4i)27-s + (16.5 − 12.0i)29-s + (−65.9 − 202. i)31-s + ⋯
L(s)  = 1  + (1.17 + 0.856i)3-s + (0.149 − 0.460i)5-s + (−0.967 + 0.702i)7-s + (0.347 + 1.07i)9-s + (0.302 + 0.953i)11-s + (0.506 + 1.55i)13-s + (0.570 − 0.414i)15-s + (0.261 − 0.804i)17-s + (0.930 + 0.676i)19-s − 1.74·21-s − 1.29·23-s + (0.619 + 0.450i)25-s + (−0.0565 + 0.174i)27-s + (0.105 − 0.0768i)29-s + (−0.381 − 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0521 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0521 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $0.0521 - 0.998i$
Analytic conductor: \(10.3843\)
Root analytic conductor: \(3.22247\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :3/2),\ 0.0521 - 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.66942 + 1.58448i\)
\(L(\frac12)\) \(\approx\) \(1.66942 + 1.58448i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.0 - 34.7i)T \)
good3 \( 1 + (-6.12 - 4.45i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (-1.67 + 5.14i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (17.9 - 13.0i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (-23.7 - 73.0i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-18.3 + 56.3i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (-77.0 - 55.9i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 142.T + 1.21e4T^{2} \)
29 \( 1 + (-16.5 + 12.0i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (65.9 + 202. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-117. + 85.5i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (67.0 + 48.6i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 - 151.T + 7.95e4T^{2} \)
47 \( 1 + (-73.1 - 53.1i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (72.5 + 223. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-244. + 177. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (46.1 - 141. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 826.T + 3.00e5T^{2} \)
71 \( 1 + (-277. + 854. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-111. + 81.1i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-94.0 - 289. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-236. + 726. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 313.T + 7.04e5T^{2} \)
97 \( 1 + (180. + 553. i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.49946233372362149229937283374, −11.63118551379828173329239737883, −9.891067820440142634186877114445, −9.448078276634624203134548990847, −8.846920616862445225015220310026, −7.48261011957975349165592136121, −6.06417673911023042647145053032, −4.52883301627833416269927028334, −3.49902630078312899385687228865, −2.09998270452172694222598994813, 0.967777356489846957668027175201, 2.89837934027946008569923001493, 3.54442013394143753479488666645, 5.89671946498830028919707301629, 6.90373136838408278745781172840, 7.911867253816994843972669902011, 8.718124812818259711034519757652, 9.995609868348587577637053995686, 10.82381530082401071921751746927, 12.38083643622756365826736595062

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