Properties

Label 2-177-1.1-c11-0-62
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  − 71.6·2-s + 243·3-s + 3.08e3·4-s − 9.71e3·5-s − 1.74e4·6-s + 3.76e4·7-s − 7.40e4·8-s + 5.90e4·9-s + 6.95e5·10-s + 6.06e4·11-s + 7.48e5·12-s + 6.24e5·13-s − 2.69e6·14-s − 2.35e6·15-s − 1.01e6·16-s + 2.49e6·17-s − 4.22e6·18-s − 1.44e7·19-s − 2.99e7·20-s + 9.14e6·21-s − 4.34e6·22-s + 1.14e7·23-s − 1.79e7·24-s + 4.54e7·25-s − 4.47e7·26-s + 1.43e7·27-s + 1.15e8·28-s + ⋯
L(s)  = 1  − 1.58·2-s + 0.577·3-s + 1.50·4-s − 1.38·5-s − 0.913·6-s + 0.846·7-s − 0.798·8-s + 0.333·9-s + 2.19·10-s + 0.113·11-s + 0.868·12-s + 0.466·13-s − 1.33·14-s − 0.802·15-s − 0.240·16-s + 0.425·17-s − 0.527·18-s − 1.33·19-s − 2.09·20-s + 0.488·21-s − 0.179·22-s + 0.371·23-s − 0.460·24-s + 0.930·25-s − 0.738·26-s + 0.192·27-s + 1.27·28-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 \( 1 + 71.6T + 2.04e3T^{2} \)
5 \( 1 + 9.71e3T + 4.88e7T^{2} \)
7 \( 1 - 3.76e4T + 1.97e9T^{2} \)
11 \( 1 - 6.06e4T + 2.85e11T^{2} \)
13 \( 1 - 6.24e5T + 1.79e12T^{2} \)
17 \( 1 - 2.49e6T + 3.42e13T^{2} \)
19 \( 1 + 1.44e7T + 1.16e14T^{2} \)
23 \( 1 - 1.14e7T + 9.52e14T^{2} \)
29 \( 1 + 1.23e8T + 1.22e16T^{2} \)
31 \( 1 - 2.37e8T + 2.54e16T^{2} \)
37 \( 1 + 4.08e8T + 1.77e17T^{2} \)
41 \( 1 + 2.82e8T + 5.50e17T^{2} \)
43 \( 1 - 1.04e9T + 9.29e17T^{2} \)
47 \( 1 + 2.58e9T + 2.47e18T^{2} \)
53 \( 1 + 1.74e9T + 9.26e18T^{2} \)
61 \( 1 - 1.03e10T + 4.35e19T^{2} \)
67 \( 1 - 1.41e10T + 1.22e20T^{2} \)
71 \( 1 + 1.38e10T + 2.31e20T^{2} \)
73 \( 1 - 1.71e10T + 3.13e20T^{2} \)
79 \( 1 - 1.65e10T + 7.47e20T^{2} \)
83 \( 1 - 1.81e10T + 1.28e21T^{2} \)
89 \( 1 + 9.62e10T + 2.77e21T^{2} \)
97 \( 1 - 6.76e10T + 7.15e21T^{2} \)
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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

−10.04497785569903084312325530169, −8.819074644448456503272647738308, −8.235370247314569116146618880980, −7.68286947316877769775913461780, −6.65184937482923644982995427838, −4.66058599840317655390336569402, −3.57001865702266185519905809794, −2.10908029425692916090467398804, −1.06096994702472619703425245188, 0, 1.06096994702472619703425245188, 2.10908029425692916090467398804, 3.57001865702266185519905809794, 4.66058599840317655390336569402, 6.65184937482923644982995427838, 7.68286947316877769775913461780, 8.235370247314569116146618880980, 8.819074644448456503272647738308, 10.04497785569903084312325530169

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