Properties

Label 2-177-177.176-c3-0-0
Degree $2$
Conductor $177$
Sign $0.213 - 0.976i$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.07 − 1.11i)3-s − 8·4-s − 22.0i·5-s − 34.4·7-s + (24.5 + 11.2i)9-s + (40.6 + 8.88i)12-s + (−24.4 + 111. i)15-s + 64·16-s − 61.4i·17-s + 40.2·19-s + 176. i·20-s + (174. + 38.2i)21-s − 360.·25-s + (−111. − 84.4i)27-s + 275.·28-s + 272. i·29-s + ⋯
L(s)  = 1  + (−0.976 − 0.213i)3-s − 4-s − 1.97i·5-s − 1.86·7-s + (0.908 + 0.417i)9-s + (0.976 + 0.213i)12-s + (−0.421 + 1.92i)15-s + 16-s − 0.876i·17-s + 0.486·19-s + 1.97i·20-s + (1.81 + 0.397i)21-s − 2.88·25-s + (−0.798 − 0.602i)27-s + 1.86·28-s + 1.74i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.213 - 0.976i$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 0.213 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0207153 + 0.0166723i\)
\(L(\frac12)\) \(\approx\) \(0.0207153 + 0.0166723i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (5.07 + 1.11i)T \)
59 \( 1 - 453. iT \)
good2 \( 1 + 8T^{2} \)
5 \( 1 + 22.0iT - 125T^{2} \)
7 \( 1 + 34.4T + 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 - 2.19e3T^{2} \)
17 \( 1 + 61.4iT - 4.91e3T^{2} \)
19 \( 1 - 40.2T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 - 272. iT - 2.43e4T^{2} \)
31 \( 1 - 2.97e4T^{2} \)
37 \( 1 - 5.06e4T^{2} \)
41 \( 1 + 458. iT - 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 632. iT - 1.48e5T^{2} \)
61 \( 1 - 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 - 1.18e3iT - 3.57e5T^{2} \)
73 \( 1 - 3.89e5T^{2} \)
79 \( 1 + 560.T + 4.93e5T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 - 9.12e5T^{2} \)
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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

−12.64540613675810762197334675579, −11.91840140978515393363573127629, −10.13460718286402337594242598697, −9.416521741846043842526257274450, −8.703130671531721083388515498000, −7.16064898845484828680573886858, −5.69731312289455907248978754614, −5.04493462268975431818553505039, −3.83200236561842820092989515820, −0.914810407908609963011206365389, 0.01938114876531591030430636149, 3.10488000042362624741191071477, 4.03510180718320369584739869492, 5.98382194852621079607289776790, 6.40171858580779191946942758084, 7.59428558578787852750538549269, 9.583480630937734595571971488649, 9.986640743766893664643289831022, 10.81477019405885453773852343128, 11.98762802596587791184058429882

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