Properties

Label 2-177-1.1-c7-0-50
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 22.3·2-s + 27·3-s + 370.·4-s − 84.4·5-s − 602.·6-s + 1.53e3·7-s − 5.40e3·8-s + 729·9-s + 1.88e3·10-s + 4.63e3·11-s + 9.99e3·12-s − 1.71e3·13-s − 3.41e4·14-s − 2.27e3·15-s + 7.32e4·16-s − 1.42e4·17-s − 1.62e4·18-s − 2.19e4·19-s − 3.12e4·20-s + 4.13e4·21-s − 1.03e5·22-s − 1.07e5·23-s − 1.45e5·24-s − 7.09e4·25-s + 3.81e4·26-s + 1.96e4·27-s + 5.67e5·28-s + ⋯
L(s)  = 1  − 1.97·2-s + 0.577·3-s + 2.89·4-s − 0.302·5-s − 1.13·6-s + 1.68·7-s − 3.73·8-s + 0.333·9-s + 0.595·10-s + 1.05·11-s + 1.66·12-s − 0.215·13-s − 3.33·14-s − 0.174·15-s + 4.46·16-s − 0.702·17-s − 0.657·18-s − 0.732·19-s − 0.873·20-s + 0.974·21-s − 2.07·22-s − 1.84·23-s − 2.15·24-s − 0.908·25-s + 0.425·26-s + 0.192·27-s + 4.88·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 + 2.05e5T \)
good2 \( 1 + 22.3T + 128T^{2} \)
5 \( 1 + 84.4T + 7.81e4T^{2} \)
7 \( 1 - 1.53e3T + 8.23e5T^{2} \)
11 \( 1 - 4.63e3T + 1.94e7T^{2} \)
13 \( 1 + 1.71e3T + 6.27e7T^{2} \)
17 \( 1 + 1.42e4T + 4.10e8T^{2} \)
19 \( 1 + 2.19e4T + 8.93e8T^{2} \)
23 \( 1 + 1.07e5T + 3.40e9T^{2} \)
29 \( 1 + 1.95e5T + 1.72e10T^{2} \)
31 \( 1 + 6.81e4T + 2.75e10T^{2} \)
37 \( 1 + 3.09e5T + 9.49e10T^{2} \)
41 \( 1 - 8.73e5T + 1.94e11T^{2} \)
43 \( 1 - 1.59e5T + 2.71e11T^{2} \)
47 \( 1 + 6.15e5T + 5.06e11T^{2} \)
53 \( 1 + 7.45e5T + 1.17e12T^{2} \)
61 \( 1 + 2.69e6T + 3.14e12T^{2} \)
67 \( 1 - 2.11e6T + 6.06e12T^{2} \)
71 \( 1 + 1.64e6T + 9.09e12T^{2} \)
73 \( 1 + 2.91e6T + 1.10e13T^{2} \)
79 \( 1 - 1.71e5T + 1.92e13T^{2} \)
83 \( 1 - 4.94e6T + 2.71e13T^{2} \)
89 \( 1 + 1.83e6T + 4.42e13T^{2} \)
97 \( 1 + 1.09e7T + 8.07e13T^{2} \)
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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

−10.81315520107305850988290855341, −9.594225500661542290429045501200, −8.796745723437036379863819896493, −7.992930088276798851762436056205, −7.40331594173308338109950920329, −6.06926217721359234497877675265, −4.00949575850272207824771197436, −2.10374308582576415374665622899, −1.57646184492538192657996692310, 0, 1.57646184492538192657996692310, 2.10374308582576415374665622899, 4.00949575850272207824771197436, 6.06926217721359234497877675265, 7.40331594173308338109950920329, 7.992930088276798851762436056205, 8.796745723437036379863819896493, 9.594225500661542290429045501200, 10.81315520107305850988290855341

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