Properties

Label 2-177-1.1-c7-0-22
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  − 22.1·2-s − 27·3-s + 363.·4-s + 258.·5-s + 598.·6-s + 411.·7-s − 5.21e3·8-s + 729·9-s − 5.73e3·10-s + 437.·11-s − 9.81e3·12-s + 1.32e4·13-s − 9.11e3·14-s − 6.98e3·15-s + 6.91e4·16-s + 2.49e4·17-s − 1.61e4·18-s + 3.80e4·19-s + 9.40e4·20-s − 1.10e4·21-s − 9.70e3·22-s + 7.97e4·23-s + 1.40e5·24-s − 1.11e4·25-s − 2.93e5·26-s − 1.96e4·27-s + 1.49e5·28-s + ⋯
L(s)  = 1  − 1.95·2-s − 0.577·3-s + 2.83·4-s + 0.925·5-s + 1.13·6-s + 0.452·7-s − 3.60·8-s + 0.333·9-s − 1.81·10-s + 0.0991·11-s − 1.63·12-s + 1.66·13-s − 0.887·14-s − 0.534·15-s + 4.22·16-s + 1.23·17-s − 0.653·18-s + 1.27·19-s + 2.62·20-s − 0.261·21-s − 0.194·22-s + 1.36·23-s + 2.08·24-s − 0.142·25-s − 3.27·26-s − 0.192·27-s + 1.28·28-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

\(L(4)\) \(\approx\) \(1.194912097\)
\(L(\frac12)\) \(\approx\) \(1.194912097\)
\(L(\frac{9}{2})\) 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 \( 1 + 22.1T + 128T^{2} \)
5 \( 1 - 258.T + 7.81e4T^{2} \)
7 \( 1 - 411.T + 8.23e5T^{2} \)
11 \( 1 - 437.T + 1.94e7T^{2} \)
13 \( 1 - 1.32e4T + 6.27e7T^{2} \)
17 \( 1 - 2.49e4T + 4.10e8T^{2} \)
19 \( 1 - 3.80e4T + 8.93e8T^{2} \)
23 \( 1 - 7.97e4T + 3.40e9T^{2} \)
29 \( 1 + 6.43e4T + 1.72e10T^{2} \)
31 \( 1 - 2.41e5T + 2.75e10T^{2} \)
37 \( 1 - 3.53e5T + 9.49e10T^{2} \)
41 \( 1 - 2.46e5T + 1.94e11T^{2} \)
43 \( 1 + 1.51e5T + 2.71e11T^{2} \)
47 \( 1 + 5.12e5T + 5.06e11T^{2} \)
53 \( 1 + 1.41e5T + 1.17e12T^{2} \)
61 \( 1 - 8.46e5T + 3.14e12T^{2} \)
67 \( 1 - 3.36e6T + 6.06e12T^{2} \)
71 \( 1 + 4.07e6T + 9.09e12T^{2} \)
73 \( 1 + 4.42e6T + 1.10e13T^{2} \)
79 \( 1 + 4.19e6T + 1.92e13T^{2} \)
83 \( 1 + 6.96e6T + 2.71e13T^{2} \)
89 \( 1 + 2.48e6T + 4.42e13T^{2} \)
97 \( 1 - 8.97e6T + 8.07e13T^{2} \)
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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

−11.19282203263433505259301162285, −10.14661908346747413048422635561, −9.519665685055646619412107060164, −8.477140236291270288268382240331, −7.51144387459829069445425792368, −6.34327863751445251913867034157, −5.59306697739451354477431434136, −3.04703550593860676751884247026, −1.43865393110429910051739118575, −0.976296513947561666190648431919, 0.976296513947561666190648431919, 1.43865393110429910051739118575, 3.04703550593860676751884247026, 5.59306697739451354477431434136, 6.34327863751445251913867034157, 7.51144387459829069445425792368, 8.477140236291270288268382240331, 9.519665685055646619412107060164, 10.14661908346747413048422635561, 11.19282203263433505259301162285

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