Properties

Label 2-177-1.1-c7-0-20
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.49·2-s − 27·3-s − 71.8·4-s + 400.·5-s − 202.·6-s − 272.·7-s − 1.49e3·8-s + 729·9-s + 3.00e3·10-s + 1.28e3·11-s + 1.94e3·12-s + 9.53e3·13-s − 2.03e3·14-s − 1.08e4·15-s − 2.02e3·16-s − 3.54e4·17-s + 5.46e3·18-s − 1.44e4·19-s − 2.87e4·20-s + 7.34e3·21-s + 9.64e3·22-s + 5.12e4·23-s + 4.04e4·24-s + 8.25e4·25-s + 7.14e4·26-s − 1.96e4·27-s + 1.95e4·28-s + ⋯
L(s)  = 1  + 0.662·2-s − 0.577·3-s − 0.561·4-s + 1.43·5-s − 0.382·6-s − 0.299·7-s − 1.03·8-s + 0.333·9-s + 0.949·10-s + 0.291·11-s + 0.324·12-s + 1.20·13-s − 0.198·14-s − 0.827·15-s − 0.123·16-s − 1.75·17-s + 0.220·18-s − 0.482·19-s − 0.804·20-s + 0.173·21-s + 0.193·22-s + 0.878·23-s + 0.597·24-s + 1.05·25-s + 0.797·26-s − 0.192·27-s + 0.168·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(2.466589527\)
\(L(\frac12)\) \(\approx\) \(2.466589527\)
\(L(\frac{9}{2})\) 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 \( 1 - 7.49T + 128T^{2} \)
5 \( 1 - 400.T + 7.81e4T^{2} \)
7 \( 1 + 272.T + 8.23e5T^{2} \)
11 \( 1 - 1.28e3T + 1.94e7T^{2} \)
13 \( 1 - 9.53e3T + 6.27e7T^{2} \)
17 \( 1 + 3.54e4T + 4.10e8T^{2} \)
19 \( 1 + 1.44e4T + 8.93e8T^{2} \)
23 \( 1 - 5.12e4T + 3.40e9T^{2} \)
29 \( 1 + 7.99e4T + 1.72e10T^{2} \)
31 \( 1 - 2.28e5T + 2.75e10T^{2} \)
37 \( 1 + 1.39e5T + 9.49e10T^{2} \)
41 \( 1 - 4.97e5T + 1.94e11T^{2} \)
43 \( 1 - 2.99e5T + 2.71e11T^{2} \)
47 \( 1 - 2.80e5T + 5.06e11T^{2} \)
53 \( 1 - 1.78e6T + 1.17e12T^{2} \)
61 \( 1 - 3.36e6T + 3.14e12T^{2} \)
67 \( 1 + 5.85e5T + 6.06e12T^{2} \)
71 \( 1 - 2.83e6T + 9.09e12T^{2} \)
73 \( 1 + 7.58e5T + 1.10e13T^{2} \)
79 \( 1 - 3.27e6T + 1.92e13T^{2} \)
83 \( 1 - 9.92e6T + 2.71e13T^{2} \)
89 \( 1 + 6.34e5T + 4.42e13T^{2} \)
97 \( 1 - 8.12e6T + 8.07e13T^{2} \)
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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

−11.41647011541473544409214238720, −10.44868118568169819486550090982, −9.338533814886912111813732244430, −8.710697989328227669532355611143, −6.58285796874294661919208917152, −6.07173931125791098443158402337, −5.03360924188892249942286253139, −3.92479452791035364223309683037, −2.33695168707338402770400068323, −0.819779363042382687118424548634, 0.819779363042382687118424548634, 2.33695168707338402770400068323, 3.92479452791035364223309683037, 5.03360924188892249942286253139, 6.07173931125791098443158402337, 6.58285796874294661919208917152, 8.710697989328227669532355611143, 9.338533814886912111813732244430, 10.44868118568169819486550090982, 11.41647011541473544409214238720

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