Properties

Label 2-177-1.1-c7-0-30
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  + 7.43·2-s + 27·3-s − 72.6·4-s + 171.·5-s + 200.·6-s + 1.52e3·7-s − 1.49e3·8-s + 729·9-s + 1.27e3·10-s − 1.63e3·11-s − 1.96e3·12-s − 7.31e3·13-s + 1.13e4·14-s + 4.63e3·15-s − 1.79e3·16-s + 3.32e4·17-s + 5.42e3·18-s + 5.55e4·19-s − 1.24e4·20-s + 4.13e4·21-s − 1.21e4·22-s − 5.18e4·23-s − 4.03e4·24-s − 4.86e4·25-s − 5.44e4·26-s + 1.96e4·27-s − 1.11e5·28-s + ⋯
L(s)  = 1  + 0.657·2-s + 0.577·3-s − 0.567·4-s + 0.614·5-s + 0.379·6-s + 1.68·7-s − 1.03·8-s + 0.333·9-s + 0.404·10-s − 0.371·11-s − 0.327·12-s − 0.923·13-s + 1.10·14-s + 0.354·15-s − 0.109·16-s + 1.63·17-s + 0.219·18-s + 1.85·19-s − 0.349·20-s + 0.973·21-s − 0.244·22-s − 0.889·23-s − 0.595·24-s − 0.622·25-s − 0.607·26-s + 0.192·27-s − 0.957·28-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

\(L(4)\) \(\approx\) \(4.207847957\)
\(L(\frac12)\) \(\approx\) \(4.207847957\)
\(L(\frac{9}{2})\) 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 \( 1 - 7.43T + 128T^{2} \)
5 \( 1 - 171.T + 7.81e4T^{2} \)
7 \( 1 - 1.52e3T + 8.23e5T^{2} \)
11 \( 1 + 1.63e3T + 1.94e7T^{2} \)
13 \( 1 + 7.31e3T + 6.27e7T^{2} \)
17 \( 1 - 3.32e4T + 4.10e8T^{2} \)
19 \( 1 - 5.55e4T + 8.93e8T^{2} \)
23 \( 1 + 5.18e4T + 3.40e9T^{2} \)
29 \( 1 - 1.03e4T + 1.72e10T^{2} \)
31 \( 1 - 9.27e4T + 2.75e10T^{2} \)
37 \( 1 + 5.04e5T + 9.49e10T^{2} \)
41 \( 1 - 6.35e5T + 1.94e11T^{2} \)
43 \( 1 - 1.01e5T + 2.71e11T^{2} \)
47 \( 1 + 7.24e4T + 5.06e11T^{2} \)
53 \( 1 - 1.67e6T + 1.17e12T^{2} \)
61 \( 1 - 7.27e5T + 3.14e12T^{2} \)
67 \( 1 - 2.81e6T + 6.06e12T^{2} \)
71 \( 1 + 8.50e5T + 9.09e12T^{2} \)
73 \( 1 - 5.84e6T + 1.10e13T^{2} \)
79 \( 1 - 2.74e6T + 1.92e13T^{2} \)
83 \( 1 - 2.78e6T + 2.71e13T^{2} \)
89 \( 1 + 6.16e6T + 4.42e13T^{2} \)
97 \( 1 + 1.67e7T + 8.07e13T^{2} \)
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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

−11.75574945093105642794903487282, −10.16804605761049100376336113237, −9.469751809781188409840563397915, −8.193946439994408123110785898009, −7.53948008712300747630987159457, −5.51727006817827650511765086266, −5.09317714971666176219345451010, −3.76714722261506783547807684273, −2.41635015214560518330385860526, −1.08300448984409725667560059164, 1.08300448984409725667560059164, 2.41635015214560518330385860526, 3.76714722261506783547807684273, 5.09317714971666176219345451010, 5.51727006817827650511765086266, 7.53948008712300747630987159457, 8.193946439994408123110785898009, 9.469751809781188409840563397915, 10.16804605761049100376336113237, 11.75574945093105642794903487282

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