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Properties

Label 2-176-11.5-c3-0-13
Degree $2$
Conductor $176$
Sign $-0.999 - 0.0106i$
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  + (−4.57 − 3.32i)3-s + (−3.76 + 11.5i)5-s + (27.3 − 19.8i)7-s + (1.51 + 4.67i)9-s + (−34.2 + 12.4i)11-s + (−6.38 − 19.6i)13-s + (55.6 − 40.4i)15-s + (−18.5 + 57.2i)17-s + (−85.6 − 62.2i)19-s − 191.·21-s − 186.·23-s + (−18.7 − 13.6i)25-s + (−38.5 + 118. i)27-s + (−109. + 79.5i)29-s + (−27.6 − 85.2i)31-s + ⋯
L(s)  = 1  + (−0.879 − 0.639i)3-s + (−0.336 + 1.03i)5-s + (1.47 − 1.07i)7-s + (0.0562 + 0.173i)9-s + (−0.939 + 0.342i)11-s + (−0.136 − 0.419i)13-s + (0.957 − 0.695i)15-s + (−0.265 + 0.816i)17-s + (−1.03 − 0.751i)19-s − 1.98·21-s − 1.69·23-s + (−0.150 − 0.109i)25-s + (−0.274 + 0.845i)27-s + (−0.701 + 0.509i)29-s + (−0.160 − 0.493i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0106i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.999 - 0.0106i$
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: \((2,\ 176,\ (\ :3/2),\ -0.999 - 0.0106i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00132232 + 0.249186i\)
\(L(\frac12)\) \(\approx\) \(0.00132232 + 0.249186i\)
\(L(\frac{5}{2})\) 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 \( 1 + (34.2 - 12.4i)T \)
good3 \( 1 + (4.57 + 3.32i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (3.76 - 11.5i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (-27.3 + 19.8i)T + (105. - 326. i)T^{2} \)
13 \( 1 + (6.38 + 19.6i)T + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (18.5 - 57.2i)T + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (85.6 + 62.2i)T + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 186.T + 1.21e4T^{2} \)
29 \( 1 + (109. - 79.5i)T + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (27.6 + 85.2i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-148. + 108. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (301. + 219. i)T + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 294.T + 7.95e4T^{2} \)
47 \( 1 + (-6.86 - 4.98i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (67.9 + 209. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-210. + 153. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (159. - 489. i)T + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 - 263.T + 3.00e5T^{2} \)
71 \( 1 + (-95.6 + 294. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (-170. + 123. i)T + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-288. - 888. i)T + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-73.5 + 226. i)T + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 + 620.T + 7.04e5T^{2} \)
97 \( 1 + (-372. - 1.14e3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−11.50550498997867255725476079560, −10.88561996153039071849588366882, −10.30546499645863582723192352250, −8.231667631899586653630621681906, −7.45343275366247768634534970820, −6.61681852618210349103389064043, −5.29426997307019655591971296924, −3.96329575176438314996573478749, −1.95507604698753929826131991037, −0.12000654947139637312823181128, 2.01790990768231636747848030437, 4.45549828142160470748230147866, 5.06227873909654785406083896311, 5.92881125737313697680724689710, 8.042335243601932589139999082792, 8.459994307029439114646982497483, 9.832865660200632955679737564190, 10.99981574504129431537296045565, 11.71395572091987843306763270432, 12.34145287782639887001010733077

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