Properties

Label 2-177-1.1-c7-0-32
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  + 21.5·2-s − 27·3-s + 337.·4-s − 113.·5-s − 582.·6-s − 647.·7-s + 4.53e3·8-s + 729·9-s − 2.44e3·10-s + 797.·11-s − 9.12e3·12-s + 863.·13-s − 1.39e4·14-s + 3.06e3·15-s + 5.45e4·16-s + 3.02e4·17-s + 1.57e4·18-s + 3.66e4·19-s − 3.82e4·20-s + 1.74e4·21-s + 1.72e4·22-s + 5.50e4·23-s − 1.22e5·24-s − 6.52e4·25-s + 1.86e4·26-s − 1.96e4·27-s − 2.18e5·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.577·3-s + 2.63·4-s − 0.405·5-s − 1.10·6-s − 0.713·7-s + 3.12·8-s + 0.333·9-s − 0.773·10-s + 0.180·11-s − 1.52·12-s + 0.108·13-s − 1.36·14-s + 0.234·15-s + 3.32·16-s + 1.49·17-s + 0.635·18-s + 1.22·19-s − 1.07·20-s + 0.411·21-s + 0.344·22-s + 0.943·23-s − 1.80·24-s − 0.835·25-s + 0.207·26-s − 0.192·27-s − 1.88·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(6.064162177\)
\(L(\frac12)\) \(\approx\) \(6.064162177\)
\(L(\frac{9}{2})\) 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 \( 1 - 21.5T + 128T^{2} \)
5 \( 1 + 113.T + 7.81e4T^{2} \)
7 \( 1 + 647.T + 8.23e5T^{2} \)
11 \( 1 - 797.T + 1.94e7T^{2} \)
13 \( 1 - 863.T + 6.27e7T^{2} \)
17 \( 1 - 3.02e4T + 4.10e8T^{2} \)
19 \( 1 - 3.66e4T + 8.93e8T^{2} \)
23 \( 1 - 5.50e4T + 3.40e9T^{2} \)
29 \( 1 - 2.01e5T + 1.72e10T^{2} \)
31 \( 1 + 1.73e4T + 2.75e10T^{2} \)
37 \( 1 + 1.57e5T + 9.49e10T^{2} \)
41 \( 1 - 2.64e5T + 1.94e11T^{2} \)
43 \( 1 - 9.75e4T + 2.71e11T^{2} \)
47 \( 1 - 2.36e5T + 5.06e11T^{2} \)
53 \( 1 + 5.24e5T + 1.17e12T^{2} \)
61 \( 1 - 2.50e6T + 3.14e12T^{2} \)
67 \( 1 + 1.86e6T + 6.06e12T^{2} \)
71 \( 1 + 3.17e6T + 9.09e12T^{2} \)
73 \( 1 - 3.03e5T + 1.10e13T^{2} \)
79 \( 1 - 3.54e6T + 1.92e13T^{2} \)
83 \( 1 - 9.87e6T + 2.71e13T^{2} \)
89 \( 1 + 6.49e6T + 4.42e13T^{2} \)
97 \( 1 + 8.43e3T + 8.07e13T^{2} \)
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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

−11.91867851946964073275551470326, −10.86935257020048245076956730656, −9.807040398176565081342600492688, −7.72223223156416747621079537704, −6.79627027323523579979662981727, −5.82807373151716996450285891757, −4.98227271188145921421997412093, −3.75185017922542774215246434848, −2.93515461877849952250068418004, −1.13160060623441683884505152388, 1.13160060623441683884505152388, 2.93515461877849952250068418004, 3.75185017922542774215246434848, 4.98227271188145921421997412093, 5.82807373151716996450285891757, 6.79627027323523579979662981727, 7.72223223156416747621079537704, 9.807040398176565081342600492688, 10.86935257020048245076956730656, 11.91867851946964073275551470326

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