Properties

Label 2-177-1.1-c11-0-82
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 60.2·2-s − 243·3-s + 1.57e3·4-s − 1.05e4·5-s − 1.46e4·6-s + 7.22e4·7-s − 2.82e4·8-s + 5.90e4·9-s − 6.37e5·10-s + 2.71e5·11-s − 3.83e5·12-s − 5.46e5·13-s + 4.35e6·14-s + 2.57e6·15-s − 4.93e6·16-s + 1.75e4·17-s + 3.55e6·18-s − 1.54e6·19-s − 1.67e7·20-s − 1.75e7·21-s + 1.63e7·22-s + 2.86e7·23-s + 6.87e6·24-s + 6.31e7·25-s − 3.29e7·26-s − 1.43e7·27-s + 1.14e8·28-s + ⋯
L(s)  = 1  + 1.33·2-s − 0.577·3-s + 0.770·4-s − 1.51·5-s − 0.768·6-s + 1.62·7-s − 0.305·8-s + 0.333·9-s − 2.01·10-s + 0.508·11-s − 0.444·12-s − 0.408·13-s + 2.16·14-s + 0.874·15-s − 1.17·16-s + 0.00299·17-s + 0.443·18-s − 0.143·19-s − 1.16·20-s − 0.938·21-s + 0.677·22-s + 0.928·23-s + 0.176·24-s + 1.29·25-s − 0.543·26-s − 0.192·27-s + 1.25·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 - 60.2T + 2.04e3T^{2} \)
5 \( 1 + 1.05e4T + 4.88e7T^{2} \)
7 \( 1 - 7.22e4T + 1.97e9T^{2} \)
11 \( 1 - 2.71e5T + 2.85e11T^{2} \)
13 \( 1 + 5.46e5T + 1.79e12T^{2} \)
17 \( 1 - 1.75e4T + 3.42e13T^{2} \)
19 \( 1 + 1.54e6T + 1.16e14T^{2} \)
23 \( 1 - 2.86e7T + 9.52e14T^{2} \)
29 \( 1 - 1.81e8T + 1.22e16T^{2} \)
31 \( 1 + 2.76e8T + 2.54e16T^{2} \)
37 \( 1 - 4.29e8T + 1.77e17T^{2} \)
41 \( 1 - 7.55e7T + 5.50e17T^{2} \)
43 \( 1 - 1.28e9T + 9.29e17T^{2} \)
47 \( 1 + 4.08e8T + 2.47e18T^{2} \)
53 \( 1 + 5.10e9T + 9.26e18T^{2} \)
61 \( 1 + 5.06e9T + 4.35e19T^{2} \)
67 \( 1 + 2.02e10T + 1.22e20T^{2} \)
71 \( 1 + 5.57e9T + 2.31e20T^{2} \)
73 \( 1 + 1.58e10T + 3.13e20T^{2} \)
79 \( 1 + 1.93e10T + 7.47e20T^{2} \)
83 \( 1 + 6.53e10T + 1.28e21T^{2} \)
89 \( 1 + 4.74e10T + 2.77e21T^{2} \)
97 \( 1 + 1.44e11T + 7.15e21T^{2} \)
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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

−10.90386156657525599079993483915, −8.964356587141931490539220922033, −7.86612600361385988901888736497, −6.98799517659956241928722189698, −5.62318766102639794663842840687, −4.53024190449209772585531033477, −4.31625490449160052643819382124, −2.95027248333288669758430616246, −1.31917200436912123820181697192, 0, 1.31917200436912123820181697192, 2.95027248333288669758430616246, 4.31625490449160052643819382124, 4.53024190449209772585531033477, 5.62318766102639794663842840687, 6.98799517659956241928722189698, 7.86612600361385988901888736497, 8.964356587141931490539220922033, 10.90386156657525599079993483915

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