Properties

Label 2-177-1.1-c7-0-46
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  − 17.8·2-s + 27·3-s + 192.·4-s + 255.·5-s − 483.·6-s − 1.07e3·7-s − 1.15e3·8-s + 729·9-s − 4.57e3·10-s + 4.74e3·11-s + 5.19e3·12-s + 5.72e3·13-s + 1.92e4·14-s + 6.89e3·15-s − 3.99e3·16-s − 1.28e4·17-s − 1.30e4·18-s − 5.21e4·19-s + 4.91e4·20-s − 2.89e4·21-s − 8.48e4·22-s − 9.82e3·23-s − 3.11e4·24-s − 1.29e4·25-s − 1.02e5·26-s + 1.96e4·27-s − 2.06e5·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 0.577·3-s + 1.50·4-s + 0.913·5-s − 0.913·6-s − 1.18·7-s − 0.796·8-s + 0.333·9-s − 1.44·10-s + 1.07·11-s + 0.867·12-s + 0.722·13-s + 1.87·14-s + 0.527·15-s − 0.243·16-s − 0.634·17-s − 0.527·18-s − 1.74·19-s + 1.37·20-s − 0.682·21-s − 1.69·22-s − 0.168·23-s − 0.459·24-s − 0.165·25-s − 1.14·26-s + 0.192·27-s − 1.77·28-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 \( 1 + 17.8T + 128T^{2} \)
5 \( 1 - 255.T + 7.81e4T^{2} \)
7 \( 1 + 1.07e3T + 8.23e5T^{2} \)
11 \( 1 - 4.74e3T + 1.94e7T^{2} \)
13 \( 1 - 5.72e3T + 6.27e7T^{2} \)
17 \( 1 + 1.28e4T + 4.10e8T^{2} \)
19 \( 1 + 5.21e4T + 8.93e8T^{2} \)
23 \( 1 + 9.82e3T + 3.40e9T^{2} \)
29 \( 1 + 1.96e5T + 1.72e10T^{2} \)
31 \( 1 - 2.72e5T + 2.75e10T^{2} \)
37 \( 1 - 3.09e5T + 9.49e10T^{2} \)
41 \( 1 + 4.21e5T + 1.94e11T^{2} \)
43 \( 1 + 4.06e5T + 2.71e11T^{2} \)
47 \( 1 + 2.61e5T + 5.06e11T^{2} \)
53 \( 1 - 7.38e5T + 1.17e12T^{2} \)
61 \( 1 - 2.33e5T + 3.14e12T^{2} \)
67 \( 1 - 3.54e6T + 6.06e12T^{2} \)
71 \( 1 - 1.03e6T + 9.09e12T^{2} \)
73 \( 1 + 5.88e6T + 1.10e13T^{2} \)
79 \( 1 - 4.34e6T + 1.92e13T^{2} \)
83 \( 1 + 4.12e6T + 2.71e13T^{2} \)
89 \( 1 + 7.32e6T + 4.42e13T^{2} \)
97 \( 1 - 1.46e7T + 8.07e13T^{2} \)
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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

−10.37403316232408012212469337930, −9.666811965923686830076640152606, −9.024267063745054654415431696807, −8.241724768029918615095910680899, −6.70585947019496546490481513452, −6.28588270310964493559292572626, −3.97840215011889821406717030024, −2.42315543647055390817202467667, −1.44268089018856436986875847428, 0, 1.44268089018856436986875847428, 2.42315543647055390817202467667, 3.97840215011889821406717030024, 6.28588270310964493559292572626, 6.70585947019496546490481513452, 8.241724768029918615095910680899, 9.024267063745054654415431696807, 9.666811965923686830076640152606, 10.37403316232408012212469337930

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