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Properties

Label 2-177-59.58-c6-0-52
Degree $2$
Conductor $177$
Sign $-0.854 + 0.518i$
Analytic cond. $40.7195$
Root an. cond. $6.38118$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.91i·2-s + 15.5·3-s − 15.4·4-s + 175.·5-s − 138. i·6-s − 256.·7-s − 432. i·8-s + 243·9-s − 1.56e3i·10-s − 678. i·11-s − 241.·12-s − 219. i·13-s + 2.28e3i·14-s + 2.73e3·15-s − 4.84e3·16-s + 1.38e3·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + 0.577·3-s − 0.241·4-s + 1.40·5-s − 0.643i·6-s − 0.746·7-s − 0.844i·8-s + 0.333·9-s − 1.56i·10-s − 0.509i·11-s − 0.139·12-s − 0.0998i·13-s + 0.832i·14-s + 0.811·15-s − 1.18·16-s + 0.281·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.854 + 0.518i$
Analytic conductor: \(40.7195\)
Root analytic conductor: \(6.38118\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3),\ -0.854 + 0.518i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.037392994\)
\(L(\frac12)\) \(\approx\) \(3.037392994\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 15.5T \)
59 \( 1 + (1.75e5 - 1.06e5i)T \)
good2 \( 1 + 8.91iT - 64T^{2} \)
5 \( 1 - 175.T + 1.56e4T^{2} \)
7 \( 1 + 256.T + 1.17e5T^{2} \)
11 \( 1 + 678. iT - 1.77e6T^{2} \)
13 \( 1 + 219. iT - 4.82e6T^{2} \)
17 \( 1 - 1.38e3T + 2.41e7T^{2} \)
19 \( 1 + 9.22e3T + 4.70e7T^{2} \)
23 \( 1 + 1.43e4iT - 1.48e8T^{2} \)
29 \( 1 - 3.43e4T + 5.94e8T^{2} \)
31 \( 1 + 2.26e4iT - 8.87e8T^{2} \)
37 \( 1 + 5.08e4iT - 2.56e9T^{2} \)
41 \( 1 - 1.11e5T + 4.75e9T^{2} \)
43 \( 1 + 5.49e3iT - 6.32e9T^{2} \)
47 \( 1 + 1.14e5iT - 1.07e10T^{2} \)
53 \( 1 - 7.49e4T + 2.21e10T^{2} \)
61 \( 1 + 1.90e5iT - 5.15e10T^{2} \)
67 \( 1 - 4.21e5iT - 9.04e10T^{2} \)
71 \( 1 + 3.52e4T + 1.28e11T^{2} \)
73 \( 1 - 6.98e5iT - 1.51e11T^{2} \)
79 \( 1 - 2.63e5T + 2.43e11T^{2} \)
83 \( 1 - 2.51e5iT - 3.26e11T^{2} \)
89 \( 1 - 4.29e5iT - 4.96e11T^{2} \)
97 \( 1 - 1.98e5iT - 8.32e11T^{2} \)
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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

−10.91233458101393692261479966364, −10.19488477252897581376325918152, −9.508687491613164239404736426418, −8.536421635494668789074655520665, −6.77640490060800251548725682093, −5.95737765428181661501900446829, −4.15085443220955567431246058844, −2.79743126882829415583281176455, −2.15103889351766023727445381823, −0.74075124670167349189888783684, 1.72814131531231972188915633333, 2.85419381607925833308550394609, 4.75176978370606353585883905415, 6.04298931490817550608768384344, 6.60038526332646559457592071971, 7.76697359307471278164549182639, 8.934539291275088503797205753823, 9.713128674749917926406724518387, 10.68102226151184056096457589470, 12.32451585523733207618842728184

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