Properties

Label 2-177-1.1-c7-0-36
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.85·2-s + 27·3-s − 81.0·4-s − 190.·5-s − 185.·6-s − 799.·7-s + 1.43e3·8-s + 729·9-s + 1.30e3·10-s + 972.·11-s − 2.18e3·12-s − 1.27e3·13-s + 5.47e3·14-s − 5.14e3·15-s + 552.·16-s + 3.64e4·17-s − 4.99e3·18-s + 1.90e3·19-s + 1.54e4·20-s − 2.15e4·21-s − 6.66e3·22-s + 8.01e4·23-s + 3.86e4·24-s − 4.17e4·25-s + 8.76e3·26-s + 1.96e4·27-s + 6.47e4·28-s + ⋯
L(s)  = 1  − 0.605·2-s + 0.577·3-s − 0.633·4-s − 0.682·5-s − 0.349·6-s − 0.880·7-s + 0.989·8-s + 0.333·9-s + 0.413·10-s + 0.220·11-s − 0.365·12-s − 0.161·13-s + 0.533·14-s − 0.393·15-s + 0.0337·16-s + 1.80·17-s − 0.201·18-s + 0.0635·19-s + 0.431·20-s − 0.508·21-s − 0.133·22-s + 1.37·23-s + 0.571·24-s − 0.534·25-s + 0.0977·26-s + 0.192·27-s + 0.557·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 6.85T + 128T^{2} \)
5 \( 1 + 190.T + 7.81e4T^{2} \)
7 \( 1 + 799.T + 8.23e5T^{2} \)
11 \( 1 - 972.T + 1.94e7T^{2} \)
13 \( 1 + 1.27e3T + 6.27e7T^{2} \)
17 \( 1 - 3.64e4T + 4.10e8T^{2} \)
19 \( 1 - 1.90e3T + 8.93e8T^{2} \)
23 \( 1 - 8.01e4T + 3.40e9T^{2} \)
29 \( 1 + 2.01e5T + 1.72e10T^{2} \)
31 \( 1 - 1.06e5T + 2.75e10T^{2} \)
37 \( 1 + 2.46e5T + 9.49e10T^{2} \)
41 \( 1 + 3.48e5T + 1.94e11T^{2} \)
43 \( 1 - 3.03e5T + 2.71e11T^{2} \)
47 \( 1 - 1.12e6T + 5.06e11T^{2} \)
53 \( 1 - 5.34e5T + 1.17e12T^{2} \)
61 \( 1 + 2.14e6T + 3.14e12T^{2} \)
67 \( 1 + 1.58e5T + 6.06e12T^{2} \)
71 \( 1 + 3.94e6T + 9.09e12T^{2} \)
73 \( 1 - 3.25e6T + 1.10e13T^{2} \)
79 \( 1 + 5.28e6T + 1.92e13T^{2} \)
83 \( 1 + 3.63e6T + 2.71e13T^{2} \)
89 \( 1 + 1.34e6T + 4.42e13T^{2} \)
97 \( 1 - 2.79e6T + 8.07e13T^{2} \)
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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

−10.55828061132309853123009921489, −9.664717350874510820921766033493, −8.975406639596639790750209958134, −7.88419937971395501992811133594, −7.17259816703155501266877967111, −5.47138776526515267701097221003, −4.03040479119155384552197258251, −3.15167367382176200492855822502, −1.23701456711141454853361879988, 0, 1.23701456711141454853361879988, 3.15167367382176200492855822502, 4.03040479119155384552197258251, 5.47138776526515267701097221003, 7.17259816703155501266877967111, 7.88419937971395501992811133594, 8.975406639596639790750209958134, 9.664717350874510820921766033493, 10.55828061132309853123009921489

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