Properties

Label 2-177-1.1-c3-0-18
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.26·2-s − 3·3-s + 19.7·4-s + 11.8·5-s − 15.7·6-s − 3.36·7-s + 61.6·8-s + 9·9-s + 62.5·10-s − 3.40·11-s − 59.1·12-s − 54.8·13-s − 17.7·14-s − 35.6·15-s + 166.·16-s + 68.7·17-s + 47.3·18-s + 8.88·19-s + 234.·20-s + 10.0·21-s − 17.9·22-s + 80.8·23-s − 184.·24-s + 16.1·25-s − 288.·26-s − 27·27-s − 66.3·28-s + ⋯
L(s)  = 1  + 1.86·2-s − 0.577·3-s + 2.46·4-s + 1.06·5-s − 1.07·6-s − 0.181·7-s + 2.72·8-s + 0.333·9-s + 1.97·10-s − 0.0932·11-s − 1.42·12-s − 1.17·13-s − 0.338·14-s − 0.613·15-s + 2.60·16-s + 0.980·17-s + 0.620·18-s + 0.107·19-s + 2.61·20-s + 0.104·21-s − 0.173·22-s + 0.732·23-s − 1.57·24-s + 0.129·25-s − 2.17·26-s − 0.192·27-s − 0.447·28-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.857878983\)
\(L(\frac12)\) \(\approx\) \(4.857878983\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
59 \( 1 + 59T \)
good2 \( 1 - 5.26T + 8T^{2} \)
5 \( 1 - 11.8T + 125T^{2} \)
7 \( 1 + 3.36T + 343T^{2} \)
11 \( 1 + 3.40T + 1.33e3T^{2} \)
13 \( 1 + 54.8T + 2.19e3T^{2} \)
17 \( 1 - 68.7T + 4.91e3T^{2} \)
19 \( 1 - 8.88T + 6.85e3T^{2} \)
23 \( 1 - 80.8T + 1.21e4T^{2} \)
29 \( 1 + 235.T + 2.43e4T^{2} \)
31 \( 1 - 145.T + 2.97e4T^{2} \)
37 \( 1 + 309.T + 5.06e4T^{2} \)
41 \( 1 - 37.6T + 6.89e4T^{2} \)
43 \( 1 + 465.T + 7.95e4T^{2} \)
47 \( 1 - 271.T + 1.03e5T^{2} \)
53 \( 1 + 82.0T + 1.48e5T^{2} \)
61 \( 1 + 736.T + 2.26e5T^{2} \)
67 \( 1 - 768.T + 3.00e5T^{2} \)
71 \( 1 + 164.T + 3.57e5T^{2} \)
73 \( 1 - 445.T + 3.89e5T^{2} \)
79 \( 1 - 602.T + 4.93e5T^{2} \)
83 \( 1 + 779.T + 5.71e5T^{2} \)
89 \( 1 - 1.39e3T + 7.04e5T^{2} \)
97 \( 1 + 269.T + 9.12e5T^{2} \)
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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

−12.43642218462910619726380330768, −11.67225244959342003165816301135, −10.53590677684481881085395359166, −9.652020898345426902958275931313, −7.47246303112744110486387561006, −6.48551601027429180692333594700, −5.52150165195575619991365254423, −4.89028542887256936587192330219, −3.30402882192459559539321681417, −1.92412853782076450916311592169, 1.92412853782076450916311592169, 3.30402882192459559539321681417, 4.89028542887256936587192330219, 5.52150165195575619991365254423, 6.48551601027429180692333594700, 7.47246303112744110486387561006, 9.652020898345426902958275931313, 10.53590677684481881085395359166, 11.67225244959342003165816301135, 12.43642218462910619726380330768

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