Properties

Label 2-177-1.1-c7-0-51
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  + 0.396·2-s + 27·3-s − 127.·4-s + 247.·5-s + 10.7·6-s − 652.·7-s − 101.·8-s + 729·9-s + 98.1·10-s − 2.47e3·11-s − 3.45e3·12-s + 9.41e3·13-s − 259.·14-s + 6.67e3·15-s + 1.63e4·16-s + 1.25e4·17-s + 289.·18-s − 4.13e4·19-s − 3.16e4·20-s − 1.76e4·21-s − 983.·22-s − 5.97e4·23-s − 2.74e3·24-s − 1.70e4·25-s + 3.73e3·26-s + 1.96e4·27-s + 8.34e4·28-s + ⋯
L(s)  = 1  + 0.0350·2-s + 0.577·3-s − 0.998·4-s + 0.884·5-s + 0.0202·6-s − 0.719·7-s − 0.0701·8-s + 0.333·9-s + 0.0310·10-s − 0.561·11-s − 0.576·12-s + 1.18·13-s − 0.0252·14-s + 0.510·15-s + 0.996·16-s + 0.620·17-s + 0.0116·18-s − 1.38·19-s − 0.883·20-s − 0.415·21-s − 0.0196·22-s − 1.02·23-s − 0.0404·24-s − 0.217·25-s + 0.0417·26-s + 0.192·27-s + 0.718·28-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: Trivial
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 \( 1 - 0.396T + 128T^{2} \)
5 \( 1 - 247.T + 7.81e4T^{2} \)
7 \( 1 + 652.T + 8.23e5T^{2} \)
11 \( 1 + 2.47e3T + 1.94e7T^{2} \)
13 \( 1 - 9.41e3T + 6.27e7T^{2} \)
17 \( 1 - 1.25e4T + 4.10e8T^{2} \)
19 \( 1 + 4.13e4T + 8.93e8T^{2} \)
23 \( 1 + 5.97e4T + 3.40e9T^{2} \)
29 \( 1 - 2.12e5T + 1.72e10T^{2} \)
31 \( 1 + 7.79e3T + 2.75e10T^{2} \)
37 \( 1 + 4.47e5T + 9.49e10T^{2} \)
41 \( 1 - 1.40e4T + 1.94e11T^{2} \)
43 \( 1 + 8.96e5T + 2.71e11T^{2} \)
47 \( 1 - 2.13e4T + 5.06e11T^{2} \)
53 \( 1 - 9.10e5T + 1.17e12T^{2} \)
61 \( 1 - 8.10e5T + 3.14e12T^{2} \)
67 \( 1 + 3.83e6T + 6.06e12T^{2} \)
71 \( 1 + 3.35e6T + 9.09e12T^{2} \)
73 \( 1 - 4.61e5T + 1.10e13T^{2} \)
79 \( 1 + 1.37e6T + 1.92e13T^{2} \)
83 \( 1 + 7.63e6T + 2.71e13T^{2} \)
89 \( 1 + 5.26e6T + 4.42e13T^{2} \)
97 \( 1 + 9.23e6T + 8.07e13T^{2} \)
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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

−10.35135429529233991858307727334, −9.977953598197719325186624675680, −8.790686303821147788760426793571, −8.208899116856546786223613735871, −6.51650243417015005905868689061, −5.57161710558010497553631110556, −4.18423462221505586369392273888, −3.07934233579018698658220010933, −1.58847622864308660485683554159, 0, 1.58847622864308660485683554159, 3.07934233579018698658220010933, 4.18423462221505586369392273888, 5.57161710558010497553631110556, 6.51650243417015005905868689061, 8.208899116856546786223613735871, 8.790686303821147788760426793571, 9.977953598197719325186624675680, 10.35135429529233991858307727334

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