Properties

Label 2-177-1.1-c7-0-16
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  − 5.66·2-s − 27·3-s − 95.9·4-s + 117.·5-s + 152.·6-s + 1.19e3·7-s + 1.26e3·8-s + 729·9-s − 667.·10-s − 6.53e3·11-s + 2.58e3·12-s + 1.19e4·13-s − 6.78e3·14-s − 3.18e3·15-s + 5.08e3·16-s + 2.94e4·17-s − 4.13e3·18-s + 3.45e3·19-s − 1.13e4·20-s − 3.23e4·21-s + 3.70e4·22-s − 3.77e4·23-s − 3.42e4·24-s − 6.42e4·25-s − 6.75e4·26-s − 1.96e4·27-s − 1.14e5·28-s + ⋯
L(s)  = 1  − 0.500·2-s − 0.577·3-s − 0.749·4-s + 0.421·5-s + 0.289·6-s + 1.31·7-s + 0.875·8-s + 0.333·9-s − 0.211·10-s − 1.48·11-s + 0.432·12-s + 1.50·13-s − 0.660·14-s − 0.243·15-s + 0.310·16-s + 1.45·17-s − 0.166·18-s + 0.115·19-s − 0.316·20-s − 0.761·21-s + 0.741·22-s − 0.646·23-s − 0.505·24-s − 0.822·25-s − 0.753·26-s − 0.192·27-s − 0.988·28-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

\(L(4)\) \(\approx\) \(1.322099645\)
\(L(\frac12)\) \(\approx\) \(1.322099645\)
\(L(\frac{9}{2})\) 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 \( 1 + 5.66T + 128T^{2} \)
5 \( 1 - 117.T + 7.81e4T^{2} \)
7 \( 1 - 1.19e3T + 8.23e5T^{2} \)
11 \( 1 + 6.53e3T + 1.94e7T^{2} \)
13 \( 1 - 1.19e4T + 6.27e7T^{2} \)
17 \( 1 - 2.94e4T + 4.10e8T^{2} \)
19 \( 1 - 3.45e3T + 8.93e8T^{2} \)
23 \( 1 + 3.77e4T + 3.40e9T^{2} \)
29 \( 1 + 2.07e5T + 1.72e10T^{2} \)
31 \( 1 - 1.11e5T + 2.75e10T^{2} \)
37 \( 1 + 4.95e4T + 9.49e10T^{2} \)
41 \( 1 - 1.12e5T + 1.94e11T^{2} \)
43 \( 1 + 3.89e4T + 2.71e11T^{2} \)
47 \( 1 - 1.22e6T + 5.06e11T^{2} \)
53 \( 1 + 4.75e5T + 1.17e12T^{2} \)
61 \( 1 - 6.99e5T + 3.14e12T^{2} \)
67 \( 1 + 3.23e6T + 6.06e12T^{2} \)
71 \( 1 - 5.13e6T + 9.09e12T^{2} \)
73 \( 1 - 5.08e6T + 1.10e13T^{2} \)
79 \( 1 - 7.01e6T + 1.92e13T^{2} \)
83 \( 1 - 6.20e6T + 2.71e13T^{2} \)
89 \( 1 - 1.44e6T + 4.42e13T^{2} \)
97 \( 1 + 1.19e7T + 8.07e13T^{2} \)
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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

−11.07020473280374079100413566506, −10.47249251639355704849177174099, −9.461309573297330298679536204110, −8.129193863147305498414015740597, −7.75516737259524066095596057016, −5.75638909598980089553635184417, −5.18536008529230990954923193002, −3.86776200890545219559645723108, −1.80195588374330216384607915709, −0.74490707756597320379114915980, 0.74490707756597320379114915980, 1.80195588374330216384607915709, 3.86776200890545219559645723108, 5.18536008529230990954923193002, 5.75638909598980089553635184417, 7.75516737259524066095596057016, 8.129193863147305498414015740597, 9.461309573297330298679536204110, 10.47249251639355704849177174099, 11.07020473280374079100413566506

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