Properties

Label 2-177-1.1-c11-0-87
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  − 2.16·2-s + 243·3-s − 2.04e3·4-s + 654.·5-s − 525.·6-s + 7.95e4·7-s + 8.84e3·8-s + 5.90e4·9-s − 1.41e3·10-s − 3.61e5·11-s − 4.96e5·12-s − 6.08e5·13-s − 1.71e5·14-s + 1.59e5·15-s + 4.16e6·16-s + 1.14e6·17-s − 1.27e5·18-s − 6.87e6·19-s − 1.33e6·20-s + 1.93e7·21-s + 7.82e5·22-s − 2.10e7·23-s + 2.14e6·24-s − 4.83e7·25-s + 1.31e6·26-s + 1.43e7·27-s − 1.62e8·28-s + ⋯
L(s)  = 1  − 0.0477·2-s + 0.577·3-s − 0.997·4-s + 0.0936·5-s − 0.0275·6-s + 1.78·7-s + 0.0954·8-s + 0.333·9-s − 0.00447·10-s − 0.677·11-s − 0.576·12-s − 0.454·13-s − 0.0854·14-s + 0.0540·15-s + 0.993·16-s + 0.194·17-s − 0.0159·18-s − 0.637·19-s − 0.0934·20-s + 1.03·21-s + 0.0323·22-s − 0.682·23-s + 0.0550·24-s − 0.991·25-s + 0.0217·26-s + 0.192·27-s − 1.78·28-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 \( 1 + 2.16T + 2.04e3T^{2} \)
5 \( 1 - 654.T + 4.88e7T^{2} \)
7 \( 1 - 7.95e4T + 1.97e9T^{2} \)
11 \( 1 + 3.61e5T + 2.85e11T^{2} \)
13 \( 1 + 6.08e5T + 1.79e12T^{2} \)
17 \( 1 - 1.14e6T + 3.42e13T^{2} \)
19 \( 1 + 6.87e6T + 1.16e14T^{2} \)
23 \( 1 + 2.10e7T + 9.52e14T^{2} \)
29 \( 1 + 1.41e8T + 1.22e16T^{2} \)
31 \( 1 - 8.71e7T + 2.54e16T^{2} \)
37 \( 1 - 5.08e8T + 1.77e17T^{2} \)
41 \( 1 + 2.45e8T + 5.50e17T^{2} \)
43 \( 1 - 1.63e8T + 9.29e17T^{2} \)
47 \( 1 - 1.77e9T + 2.47e18T^{2} \)
53 \( 1 + 4.11e9T + 9.26e18T^{2} \)
61 \( 1 + 2.50e9T + 4.35e19T^{2} \)
67 \( 1 + 3.37e9T + 1.22e20T^{2} \)
71 \( 1 + 6.16e9T + 2.31e20T^{2} \)
73 \( 1 - 2.20e9T + 3.13e20T^{2} \)
79 \( 1 - 2.69e10T + 7.47e20T^{2} \)
83 \( 1 + 1.69e10T + 1.28e21T^{2} \)
89 \( 1 + 5.65e10T + 2.77e21T^{2} \)
97 \( 1 + 2.14e9T + 7.15e21T^{2} \)
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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

−10.04145385729234720617347877747, −9.057655669581062452824477277293, −8.014304907823565183179481125221, −7.71804082493331435899401755163, −5.70641827773341088522036073336, −4.75242694247183180775517908961, −3.98350519171287562850784770605, −2.37331581069644556829600492318, −1.38371621742738659377305450102, 0, 1.38371621742738659377305450102, 2.37331581069644556829600492318, 3.98350519171287562850784770605, 4.75242694247183180775517908961, 5.70641827773341088522036073336, 7.71804082493331435899401755163, 8.014304907823565183179481125221, 9.057655669581062452824477277293, 10.04145385729234720617347877747

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