Properties

Label 2-177-1.1-c5-0-36
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.23·2-s + 9·3-s + 53.2·4-s + 89.5·5-s + 83.0·6-s − 121.·7-s + 195.·8-s + 81·9-s + 826.·10-s + 542.·11-s + 478.·12-s − 202.·13-s − 1.12e3·14-s + 806.·15-s + 105.·16-s − 431.·17-s + 747.·18-s + 362.·19-s + 4.76e3·20-s − 1.09e3·21-s + 5.00e3·22-s + 1.69e3·23-s + 1.76e3·24-s + 4.89e3·25-s − 1.86e3·26-s + 729·27-s − 6.46e3·28-s + ⋯
L(s)  = 1  + 1.63·2-s + 0.577·3-s + 1.66·4-s + 1.60·5-s + 0.942·6-s − 0.936·7-s + 1.08·8-s + 0.333·9-s + 2.61·10-s + 1.35·11-s + 0.960·12-s − 0.332·13-s − 1.52·14-s + 0.925·15-s + 0.102·16-s − 0.361·17-s + 0.543·18-s + 0.230·19-s + 2.66·20-s − 0.540·21-s + 2.20·22-s + 0.667·23-s + 0.624·24-s + 1.56·25-s − 0.541·26-s + 0.192·27-s − 1.55·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(7.395285689\)
\(L(\frac12)\) \(\approx\) \(7.395285689\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 + 3.48e3T \)
good2 \( 1 - 9.23T + 32T^{2} \)
5 \( 1 - 89.5T + 3.12e3T^{2} \)
7 \( 1 + 121.T + 1.68e4T^{2} \)
11 \( 1 - 542.T + 1.61e5T^{2} \)
13 \( 1 + 202.T + 3.71e5T^{2} \)
17 \( 1 + 431.T + 1.41e6T^{2} \)
19 \( 1 - 362.T + 2.47e6T^{2} \)
23 \( 1 - 1.69e3T + 6.43e6T^{2} \)
29 \( 1 + 7.67e3T + 2.05e7T^{2} \)
31 \( 1 + 2.42e3T + 2.86e7T^{2} \)
37 \( 1 - 2.53e3T + 6.93e7T^{2} \)
41 \( 1 - 6.56e3T + 1.15e8T^{2} \)
43 \( 1 - 779.T + 1.47e8T^{2} \)
47 \( 1 + 1.38e4T + 2.29e8T^{2} \)
53 \( 1 - 1.45e4T + 4.18e8T^{2} \)
61 \( 1 - 1.96e4T + 8.44e8T^{2} \)
67 \( 1 + 5.61e4T + 1.35e9T^{2} \)
71 \( 1 + 3.71e4T + 1.80e9T^{2} \)
73 \( 1 - 8.20e4T + 2.07e9T^{2} \)
79 \( 1 - 7.74e4T + 3.07e9T^{2} \)
83 \( 1 + 6.36e4T + 3.93e9T^{2} \)
89 \( 1 + 1.55e4T + 5.58e9T^{2} \)
97 \( 1 + 9.30e4T + 8.58e9T^{2} \)
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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

−12.29203123490503357480936869552, −11.03073798507435375214782482089, −9.605495224231552382208385457342, −9.178038374488707298949832676018, −7.00123663888438529834581255884, −6.28626988683504916246065834658, −5.36657180755255389582708067567, −3.97209537513429716908960495787, −2.87837749579245887162341745406, −1.75789693841219144223883818994, 1.75789693841219144223883818994, 2.87837749579245887162341745406, 3.97209537513429716908960495787, 5.36657180755255389582708067567, 6.28626988683504916246065834658, 7.00123663888438529834581255884, 9.178038374488707298949832676018, 9.605495224231552382208385457342, 11.03073798507435375214782482089, 12.29203123490503357480936869552

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