Properties

Label 2-177-59.58-c4-0-0
Degree $2$
Conductor $177$
Sign $0.486 + 0.873i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.65i·2-s − 5.19·3-s − 42.5·4-s − 38.1·5-s + 39.7i·6-s − 35.1·7-s + 203. i·8-s + 27·9-s + 292. i·10-s − 147. i·11-s + 221.·12-s − 17.0i·13-s + 268. i·14-s + 198.·15-s + 875.·16-s − 354.·17-s + ⋯
L(s)  = 1  − 1.91i·2-s − 0.577·3-s − 2.66·4-s − 1.52·5-s + 1.10i·6-s − 0.717·7-s + 3.17i·8-s + 0.333·9-s + 2.92i·10-s − 1.22i·11-s + 1.53·12-s − 0.101i·13-s + 1.37i·14-s + 0.881·15-s + 3.41·16-s − 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.09856197769\)
\(L(\frac12)\) \(\approx\) \(0.09856197769\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (-1.69e3 - 3.04e3i)T \)
good2 \( 1 + 7.65iT - 16T^{2} \)
5 \( 1 + 38.1T + 625T^{2} \)
7 \( 1 + 35.1T + 2.40e3T^{2} \)
11 \( 1 + 147. iT - 1.46e4T^{2} \)
13 \( 1 + 17.0iT - 2.85e4T^{2} \)
17 \( 1 + 354.T + 8.35e4T^{2} \)
19 \( 1 + 647.T + 1.30e5T^{2} \)
23 \( 1 + 862. iT - 2.79e5T^{2} \)
29 \( 1 - 440.T + 7.07e5T^{2} \)
31 \( 1 - 347. iT - 9.23e5T^{2} \)
37 \( 1 - 972. iT - 1.87e6T^{2} \)
41 \( 1 - 2.78e3T + 2.82e6T^{2} \)
43 \( 1 + 2.83e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.28e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.10e3T + 7.89e6T^{2} \)
61 \( 1 + 5.98e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.35e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.39e3T + 2.54e7T^{2} \)
73 \( 1 + 1.64e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.35e3T + 3.89e7T^{2} \)
83 \( 1 + 5.02e3iT - 4.74e7T^{2} \)
89 \( 1 - 5.29e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.40e4iT - 8.85e7T^{2} \)
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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

−11.71494236237603389873214149157, −10.90764525343147065685912411235, −10.50454325939921198763572830223, −8.910515088649204105796565362523, −8.307094426927261498145488282820, −6.41524183538948616930490350483, −4.58441340869422324790799367998, −3.87486461034463362682434058250, −2.66112814720322335908046778829, −0.55170050474535165824157072289, 0.084843263354554227351636252656, 4.04933073161054073073270575226, 4.60964706388521358057447568627, 6.13440977216860658335002510265, 6.99667244642392890141350159047, 7.67605860798592702514365438847, 8.768035232997542350301463405147, 9.801978116220442959265845260457, 11.25693255186467423944610836773, 12.61418737296219367138143904187

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