Properties

Label 2-177-1.1-c7-0-31
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  − 5.28·2-s + 27·3-s − 100.·4-s + 498.·5-s − 142.·6-s + 341.·7-s + 1.20e3·8-s + 729·9-s − 2.63e3·10-s + 2.09e3·11-s − 2.70e3·12-s + 4.48e3·13-s − 1.80e3·14-s + 1.34e4·15-s + 6.43e3·16-s + 2.23e4·17-s − 3.85e3·18-s + 6.30e3·19-s − 4.99e4·20-s + 9.21e3·21-s − 1.10e4·22-s + 8.94e3·23-s + 3.25e4·24-s + 1.70e5·25-s − 2.36e4·26-s + 1.96e4·27-s − 3.41e4·28-s + ⋯
L(s)  = 1  − 0.467·2-s + 0.577·3-s − 0.781·4-s + 1.78·5-s − 0.269·6-s + 0.375·7-s + 0.832·8-s + 0.333·9-s − 0.833·10-s + 0.473·11-s − 0.451·12-s + 0.565·13-s − 0.175·14-s + 1.03·15-s + 0.392·16-s + 1.10·17-s − 0.155·18-s + 0.210·19-s − 1.39·20-s + 0.217·21-s − 0.221·22-s + 0.153·23-s + 0.480·24-s + 2.18·25-s − 0.264·26-s + 0.192·27-s − 0.293·28-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

\(L(4)\) \(\approx\) \(2.889099711\)
\(L(\frac12)\) \(\approx\) \(2.889099711\)
\(L(\frac{9}{2})\) 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 \( 1 + 5.28T + 128T^{2} \)
5 \( 1 - 498.T + 7.81e4T^{2} \)
7 \( 1 - 341.T + 8.23e5T^{2} \)
11 \( 1 - 2.09e3T + 1.94e7T^{2} \)
13 \( 1 - 4.48e3T + 6.27e7T^{2} \)
17 \( 1 - 2.23e4T + 4.10e8T^{2} \)
19 \( 1 - 6.30e3T + 8.93e8T^{2} \)
23 \( 1 - 8.94e3T + 3.40e9T^{2} \)
29 \( 1 + 2.32e5T + 1.72e10T^{2} \)
31 \( 1 + 1.09e5T + 2.75e10T^{2} \)
37 \( 1 - 1.23e5T + 9.49e10T^{2} \)
41 \( 1 - 1.33e5T + 1.94e11T^{2} \)
43 \( 1 - 3.26e5T + 2.71e11T^{2} \)
47 \( 1 + 4.79e5T + 5.06e11T^{2} \)
53 \( 1 + 9.06e5T + 1.17e12T^{2} \)
61 \( 1 - 3.23e6T + 3.14e12T^{2} \)
67 \( 1 - 1.18e6T + 6.06e12T^{2} \)
71 \( 1 - 1.35e6T + 9.09e12T^{2} \)
73 \( 1 + 5.84e6T + 1.10e13T^{2} \)
79 \( 1 - 4.69e6T + 1.92e13T^{2} \)
83 \( 1 - 1.26e6T + 2.71e13T^{2} \)
89 \( 1 - 2.48e6T + 4.42e13T^{2} \)
97 \( 1 - 4.51e6T + 8.07e13T^{2} \)
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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

−11.06543861003582100877411318469, −9.935510631597059310978246516558, −9.447484255665326183534882434010, −8.647965473738288521325381458049, −7.49291549527129268815475787670, −5.99888758647539757857547162672, −5.07139619655364097496539742116, −3.56714451711836523227401839930, −1.93132062995932247802800077444, −1.09749367616250774429494856929, 1.09749367616250774429494856929, 1.93132062995932247802800077444, 3.56714451711836523227401839930, 5.07139619655364097496539742116, 5.99888758647539757857547162672, 7.49291549527129268815475787670, 8.647965473738288521325381458049, 9.447484255665326183534882434010, 9.935510631597059310978246516558, 11.06543861003582100877411318469

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