Properties

Label 2-176-176.5-c1-0-0
Degree $2$
Conductor $176$
Sign $-0.718 + 0.696i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.287 + 1.38i)2-s + (−0.925 + 0.146i)3-s + (−1.83 − 0.795i)4-s + (−1.96 + 1.00i)5-s + (0.0626 − 1.32i)6-s + (0.432 + 0.594i)7-s + (1.62 − 2.31i)8-s + (−2.01 + 0.655i)9-s + (−0.822 − 3.00i)10-s + (−1.45 − 2.97i)11-s + (1.81 + 0.466i)12-s + (−5.43 − 2.76i)13-s + (−0.947 + 0.427i)14-s + (1.67 − 1.21i)15-s + (2.73 + 2.91i)16-s + (−1.02 + 3.16i)17-s + ⋯
L(s)  = 1  + (−0.202 + 0.979i)2-s + (−0.534 + 0.0846i)3-s + (−0.917 − 0.397i)4-s + (−0.878 + 0.447i)5-s + (0.0255 − 0.540i)6-s + (0.163 + 0.224i)7-s + (0.575 − 0.817i)8-s + (−0.672 + 0.218i)9-s + (−0.260 − 0.951i)10-s + (−0.439 − 0.898i)11-s + (0.524 + 0.134i)12-s + (−1.50 − 0.768i)13-s + (−0.253 + 0.114i)14-s + (0.431 − 0.313i)15-s + (0.683 + 0.729i)16-s + (−0.249 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.718 + 0.696i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ -0.718 + 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0639036 - 0.157741i\)
\(L(\frac12)\) \(\approx\) \(0.0639036 - 0.157741i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.287 - 1.38i)T \)
11 \( 1 + (1.45 + 2.97i)T \)
good3 \( 1 + (0.925 - 0.146i)T + (2.85 - 0.927i)T^{2} \)
5 \( 1 + (1.96 - 1.00i)T + (2.93 - 4.04i)T^{2} \)
7 \( 1 + (-0.432 - 0.594i)T + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (5.43 + 2.76i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.02 - 3.16i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.56 + 0.406i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 - 8.39iT - 23T^{2} \)
29 \( 1 + (1.17 - 7.40i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (1.36 + 4.21i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.69 + 0.902i)T + (35.1 + 11.4i)T^{2} \)
41 \( 1 + (-0.420 + 0.578i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (0.0267 + 0.0267i)T + 43iT^{2} \)
47 \( 1 + (-4.68 - 3.40i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.13 - 4.18i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.3 + 1.63i)T + (56.1 + 18.2i)T^{2} \)
61 \( 1 + (-3.34 - 6.57i)T + (-35.8 + 49.3i)T^{2} \)
67 \( 1 + (-7.12 + 7.12i)T - 67iT^{2} \)
71 \( 1 + (7.44 + 2.41i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.395 + 0.544i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.81 + 5.58i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.16 - 8.17i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 16.1iT - 89T^{2} \)
97 \( 1 + (-1.32 - 4.07i)T + (-78.4 + 57.0i)T^{2} \)
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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

−13.49887026878793249666129503236, −12.28570478658540289182860040740, −11.24429699591206001458956288635, −10.40449066830654086893125730526, −9.075636437254825487835732958670, −7.922960775386963556285189267809, −7.31635867319571475011888823655, −5.78629291005349927760282431946, −5.11585743431689271447149590000, −3.38375770300743478375556179932, 0.17148681643539823150390019955, 2.52936842457867057095652768145, 4.33758956385265927505633775329, 5.06111689713385301559759441860, 7.07031539389303918113189883124, 8.108839845278908850571639132723, 9.234612858166682303855590695132, 10.22428973553318368218009211956, 11.33187240797529196280707734552, 12.15595945537333654322155024019

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