Properties

Label 2-177-1.1-c7-0-52
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  + 5.01·2-s + 27·3-s − 102.·4-s − 77.5·5-s + 135.·6-s − 216.·7-s − 1.15e3·8-s + 729·9-s − 389.·10-s + 3.37e3·11-s − 2.77e3·12-s + 1.24e4·13-s − 1.08e3·14-s − 2.09e3·15-s + 7.34e3·16-s − 2.63e4·17-s + 3.65e3·18-s + 4.14e4·19-s + 7.97e3·20-s − 5.85e3·21-s + 1.69e4·22-s − 4.21e4·23-s − 3.12e4·24-s − 7.21e4·25-s + 6.24e4·26-s + 1.96e4·27-s + 2.22e4·28-s + ⋯
L(s)  = 1  + 0.443·2-s + 0.577·3-s − 0.803·4-s − 0.277·5-s + 0.256·6-s − 0.238·7-s − 0.799·8-s + 0.333·9-s − 0.123·10-s + 0.764·11-s − 0.463·12-s + 1.57·13-s − 0.105·14-s − 0.160·15-s + 0.448·16-s − 1.30·17-s + 0.147·18-s + 1.38·19-s + 0.222·20-s − 0.137·21-s + 0.339·22-s − 0.721·23-s − 0.461·24-s − 0.922·25-s + 0.697·26-s + 0.192·27-s + 0.191·28-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 \( 1 - 5.01T + 128T^{2} \)
5 \( 1 + 77.5T + 7.81e4T^{2} \)
7 \( 1 + 216.T + 8.23e5T^{2} \)
11 \( 1 - 3.37e3T + 1.94e7T^{2} \)
13 \( 1 - 1.24e4T + 6.27e7T^{2} \)
17 \( 1 + 2.63e4T + 4.10e8T^{2} \)
19 \( 1 - 4.14e4T + 8.93e8T^{2} \)
23 \( 1 + 4.21e4T + 3.40e9T^{2} \)
29 \( 1 + 1.51e5T + 1.72e10T^{2} \)
31 \( 1 + 2.39e5T + 2.75e10T^{2} \)
37 \( 1 + 4.11e5T + 9.49e10T^{2} \)
41 \( 1 + 5.36e5T + 1.94e11T^{2} \)
43 \( 1 - 5.83e5T + 2.71e11T^{2} \)
47 \( 1 + 5.87e5T + 5.06e11T^{2} \)
53 \( 1 + 2.67e5T + 1.17e12T^{2} \)
61 \( 1 + 4.98e5T + 3.14e12T^{2} \)
67 \( 1 - 2.87e6T + 6.06e12T^{2} \)
71 \( 1 - 1.08e6T + 9.09e12T^{2} \)
73 \( 1 + 4.74e3T + 1.10e13T^{2} \)
79 \( 1 + 1.41e6T + 1.92e13T^{2} \)
83 \( 1 - 3.72e6T + 2.71e13T^{2} \)
89 \( 1 - 1.04e7T + 4.42e13T^{2} \)
97 \( 1 + 4.51e6T + 8.07e13T^{2} \)
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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

−11.02445962349741346026681472857, −9.544895332759540678233542540937, −8.969565443816026530718028024762, −7.984614648844896121320654567890, −6.59402124628909793204985325307, −5.42279456598151940850555812179, −3.94219159113592814308228928287, −3.50491298625839772572149055604, −1.60141385593376517187035292828, 0, 1.60141385593376517187035292828, 3.50491298625839772572149055604, 3.94219159113592814308228928287, 5.42279456598151940850555812179, 6.59402124628909793204985325307, 7.984614648844896121320654567890, 8.969565443816026530718028024762, 9.544895332759540678233542540937, 11.02445962349741346026681472857

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