Properties

Label 2-177-1.1-c7-0-6
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  + 11.5·2-s − 27·3-s + 6.48·4-s − 401.·5-s − 313.·6-s + 213.·7-s − 1.40e3·8-s + 729·9-s − 4.65e3·10-s − 4.97e3·11-s − 175.·12-s − 1.26e4·13-s + 2.47e3·14-s + 1.08e4·15-s − 1.71e4·16-s + 3.69e4·17-s + 8.45e3·18-s − 3.10e4·19-s − 2.60e3·20-s − 5.76e3·21-s − 5.76e4·22-s + 5.25e4·23-s + 3.80e4·24-s + 8.27e4·25-s − 1.46e5·26-s − 1.96e4·27-s + 1.38e3·28-s + ⋯
L(s)  = 1  + 1.02·2-s − 0.577·3-s + 0.0506·4-s − 1.43·5-s − 0.591·6-s + 0.235·7-s − 0.973·8-s + 0.333·9-s − 1.47·10-s − 1.12·11-s − 0.0292·12-s − 1.59·13-s + 0.241·14-s + 0.828·15-s − 1.04·16-s + 1.82·17-s + 0.341·18-s − 1.03·19-s − 0.0726·20-s − 0.135·21-s − 1.15·22-s + 0.901·23-s + 0.561·24-s + 1.05·25-s − 1.63·26-s − 0.192·27-s + 0.0119·28-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

\(L(4)\) \(\approx\) \(0.8304063611\)
\(L(\frac12)\) \(\approx\) \(0.8304063611\)
\(L(\frac{9}{2})\) 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 \( 1 - 11.5T + 128T^{2} \)
5 \( 1 + 401.T + 7.81e4T^{2} \)
7 \( 1 - 213.T + 8.23e5T^{2} \)
11 \( 1 + 4.97e3T + 1.94e7T^{2} \)
13 \( 1 + 1.26e4T + 6.27e7T^{2} \)
17 \( 1 - 3.69e4T + 4.10e8T^{2} \)
19 \( 1 + 3.10e4T + 8.93e8T^{2} \)
23 \( 1 - 5.25e4T + 3.40e9T^{2} \)
29 \( 1 + 2.09e5T + 1.72e10T^{2} \)
31 \( 1 + 1.18e4T + 2.75e10T^{2} \)
37 \( 1 - 1.69e4T + 9.49e10T^{2} \)
41 \( 1 - 8.01e5T + 1.94e11T^{2} \)
43 \( 1 - 7.20e5T + 2.71e11T^{2} \)
47 \( 1 + 5.44e4T + 5.06e11T^{2} \)
53 \( 1 + 2.95e5T + 1.17e12T^{2} \)
61 \( 1 - 1.37e6T + 3.14e12T^{2} \)
67 \( 1 - 3.05e6T + 6.06e12T^{2} \)
71 \( 1 + 4.70e6T + 9.09e12T^{2} \)
73 \( 1 + 3.65e6T + 1.10e13T^{2} \)
79 \( 1 - 6.37e6T + 1.92e13T^{2} \)
83 \( 1 + 2.08e6T + 2.71e13T^{2} \)
89 \( 1 - 1.43e6T + 4.42e13T^{2} \)
97 \( 1 + 6.42e6T + 8.07e13T^{2} \)
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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

−11.67528155969363830080808783378, −10.76659939113807925310675328721, −9.483079224408279593636831764667, −7.944066049838030854624091038566, −7.31305069405588294939367417506, −5.65369283666310383630312227544, −4.88203947208260498015619454032, −3.95617197187314381997647067994, −2.75207645751176004664817782251, −0.42055776672956387320373967960, 0.42055776672956387320373967960, 2.75207645751176004664817782251, 3.95617197187314381997647067994, 4.88203947208260498015619454032, 5.65369283666310383630312227544, 7.31305069405588294939367417506, 7.944066049838030854624091038566, 9.483079224408279593636831764667, 10.76659939113807925310675328721, 11.67528155969363830080808783378

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