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Properties

Label 2-177-3.2-c4-0-76
Degree $2$
Conductor $177$
Sign $0.859 - 0.511i$
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  − 6.25i·2-s + (−4.60 − 7.73i)3-s − 23.1·4-s − 36.4i·5-s + (−48.3 + 28.7i)6-s + 14.4·7-s + 44.4i·8-s + (−38.6 + 71.1i)9-s − 227.·10-s + 167. i·11-s + (106. + 178. i)12-s − 78.8·13-s − 90.1i·14-s + (−281. + 167. i)15-s − 91.9·16-s − 272. i·17-s + ⋯
L(s)  = 1  − 1.56i·2-s + (−0.511 − 0.859i)3-s − 1.44·4-s − 1.45i·5-s + (−1.34 + 0.799i)6-s + 0.294·7-s + 0.693i·8-s + (−0.477 + 0.878i)9-s − 2.27·10-s + 1.38i·11-s + (0.738 + 1.24i)12-s − 0.466·13-s − 0.459i·14-s + (−1.25 + 0.745i)15-s − 0.359·16-s − 0.942i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.5574201790\)
\(L(\frac12)\) \(\approx\) \(0.5574201790\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (4.60 + 7.73i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 6.25iT - 16T^{2} \)
5 \( 1 + 36.4iT - 625T^{2} \)
7 \( 1 - 14.4T + 2.40e3T^{2} \)
11 \( 1 - 167. iT - 1.46e4T^{2} \)
13 \( 1 + 78.8T + 2.85e4T^{2} \)
17 \( 1 + 272. iT - 8.35e4T^{2} \)
19 \( 1 + 177.T + 1.30e5T^{2} \)
23 \( 1 + 272. iT - 2.79e5T^{2} \)
29 \( 1 - 340. iT - 7.07e5T^{2} \)
31 \( 1 - 165.T + 9.23e5T^{2} \)
37 \( 1 - 2.33e3T + 1.87e6T^{2} \)
41 \( 1 + 1.95e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.85e3T + 3.41e6T^{2} \)
47 \( 1 + 455. iT - 4.87e6T^{2} \)
53 \( 1 + 4.27e3iT - 7.89e6T^{2} \)
61 \( 1 + 492.T + 1.38e7T^{2} \)
67 \( 1 + 3.71e3T + 2.01e7T^{2} \)
71 \( 1 - 7.64e3iT - 2.54e7T^{2} \)
73 \( 1 + 2.04e3T + 2.83e7T^{2} \)
79 \( 1 + 9.05e3T + 3.89e7T^{2} \)
83 \( 1 - 3.26e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.54e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.17e4T + 8.85e7T^{2} \)
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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

−11.52824382098763489935520673612, −10.24358557938797162669315276827, −9.332825121620275479434539818788, −8.253774439030726325583993186625, −6.96107709197335347332407243566, −5.11194718664710432516344936372, −4.47176644892199323422622271260, −2.36029407261832564772700349615, −1.38425511357407743280190612894, −0.22860012021322152838397532913, 3.14974249433429739675159404348, 4.54290746036518285467430505409, 5.98575804343751996554517052068, 6.30297881496293364171600764503, 7.62845833448624454519646675578, 8.614362028227497746568534696580, 9.879367487845308303475344425609, 10.91669459589891604015772220931, 11.53565508179457709094577118406, 13.37899417567508705351319578134

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