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Properties

Label 2-177-59.58-c4-0-1
Degree $2$
Conductor $177$
Sign $-0.669 + 0.742i$
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  + 4.64i·2-s + 5.19·3-s − 5.59·4-s − 0.691·5-s + 24.1i·6-s − 76.1·7-s + 48.3i·8-s + 27·9-s − 3.21i·10-s + 74.8i·11-s − 29.0·12-s − 105. i·13-s − 353. i·14-s − 3.59·15-s − 314.·16-s − 141.·17-s + ⋯
L(s)  = 1  + 1.16i·2-s + 0.577·3-s − 0.349·4-s − 0.0276·5-s + 0.670i·6-s − 1.55·7-s + 0.755i·8-s + 0.333·9-s − 0.0321i·10-s + 0.618i·11-s − 0.201·12-s − 0.623i·13-s − 1.80i·14-s − 0.0159·15-s − 1.22·16-s − 0.491·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.669 + 0.742i$
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: \((2,\ 177,\ (\ :2),\ -0.669 + 0.742i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.7679528127\)
\(L(\frac12)\) \(\approx\) \(0.7679528127\)
\(L(3)\) 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 \( 1 + (-2.33e3 + 2.58e3i)T \)
good2 \( 1 - 4.64iT - 16T^{2} \)
5 \( 1 + 0.691T + 625T^{2} \)
7 \( 1 + 76.1T + 2.40e3T^{2} \)
11 \( 1 - 74.8iT - 1.46e4T^{2} \)
13 \( 1 + 105. iT - 2.85e4T^{2} \)
17 \( 1 + 141.T + 8.35e4T^{2} \)
19 \( 1 + 170.T + 1.30e5T^{2} \)
23 \( 1 - 226. iT - 2.79e5T^{2} \)
29 \( 1 + 677.T + 7.07e5T^{2} \)
31 \( 1 + 114. iT - 9.23e5T^{2} \)
37 \( 1 - 488. iT - 1.87e6T^{2} \)
41 \( 1 + 1.82e3T + 2.82e6T^{2} \)
43 \( 1 - 527. iT - 3.41e6T^{2} \)
47 \( 1 + 1.90e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.68e3T + 7.89e6T^{2} \)
61 \( 1 - 6.41e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.77e3iT - 2.01e7T^{2} \)
71 \( 1 - 170.T + 2.54e7T^{2} \)
73 \( 1 - 8.99e3iT - 2.83e7T^{2} \)
79 \( 1 + 204.T + 3.89e7T^{2} \)
83 \( 1 - 1.39e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.00e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.22e3iT - 8.85e7T^{2} \)
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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

−13.04238678787172880537717960291, −11.81269519333545519381097418012, −10.31889071465336622815524537981, −9.426584461979691733529637905153, −8.395120282249582070326599153875, −7.30512703646939716941723646928, −6.56390014537255418406585811798, −5.47780355516807496918609341358, −3.82142113625191082171316353814, −2.39122469662189332705227417569, 0.23803534299371464064052544405, 2.06216475363161757530614843390, 3.20530957504131990895225989747, 4.05682261182249838764560670611, 6.17474702682957273838224477314, 7.10883268240320283527190018391, 8.754947325997453833554240271648, 9.561947976636768320316145881956, 10.35060697366268595248184207690, 11.39086990378305255807987981529

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