Properties

Label 2-177-1.1-c11-0-101
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  + 40.2·2-s + 243·3-s − 429.·4-s + 7.75e3·5-s + 9.77e3·6-s + 4.26e4·7-s − 9.96e4·8-s + 5.90e4·9-s + 3.11e5·10-s + 1.98e5·11-s − 1.04e5·12-s − 1.67e6·13-s + 1.71e6·14-s + 1.88e6·15-s − 3.13e6·16-s − 3.70e6·17-s + 2.37e6·18-s − 2.06e7·19-s − 3.32e6·20-s + 1.03e7·21-s + 7.98e6·22-s − 8.94e6·23-s − 2.42e7·24-s + 1.12e7·25-s − 6.72e7·26-s + 1.43e7·27-s − 1.82e7·28-s + ⋯
L(s)  = 1  + 0.889·2-s + 0.577·3-s − 0.209·4-s + 1.10·5-s + 0.513·6-s + 0.958·7-s − 1.07·8-s + 0.333·9-s + 0.986·10-s + 0.371·11-s − 0.120·12-s − 1.24·13-s + 0.852·14-s + 0.640·15-s − 0.746·16-s − 0.632·17-s + 0.296·18-s − 1.91·19-s − 0.232·20-s + 0.553·21-s + 0.330·22-s − 0.289·23-s − 0.620·24-s + 0.230·25-s − 1.11·26-s + 0.192·27-s − 0.200·28-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 \( 1 - 40.2T + 2.04e3T^{2} \)
5 \( 1 - 7.75e3T + 4.88e7T^{2} \)
7 \( 1 - 4.26e4T + 1.97e9T^{2} \)
11 \( 1 - 1.98e5T + 2.85e11T^{2} \)
13 \( 1 + 1.67e6T + 1.79e12T^{2} \)
17 \( 1 + 3.70e6T + 3.42e13T^{2} \)
19 \( 1 + 2.06e7T + 1.16e14T^{2} \)
23 \( 1 + 8.94e6T + 9.52e14T^{2} \)
29 \( 1 + 3.99e7T + 1.22e16T^{2} \)
31 \( 1 + 2.18e8T + 2.54e16T^{2} \)
37 \( 1 - 5.76e6T + 1.77e17T^{2} \)
41 \( 1 - 6.45e8T + 5.50e17T^{2} \)
43 \( 1 + 8.91e8T + 9.29e17T^{2} \)
47 \( 1 + 1.30e9T + 2.47e18T^{2} \)
53 \( 1 - 4.69e9T + 9.26e18T^{2} \)
61 \( 1 - 5.61e9T + 4.35e19T^{2} \)
67 \( 1 + 3.16e9T + 1.22e20T^{2} \)
71 \( 1 + 2.23e9T + 2.31e20T^{2} \)
73 \( 1 - 2.59e10T + 3.13e20T^{2} \)
79 \( 1 + 4.32e10T + 7.47e20T^{2} \)
83 \( 1 + 1.72e10T + 1.28e21T^{2} \)
89 \( 1 - 1.57e10T + 2.77e21T^{2} \)
97 \( 1 + 9.36e10T + 7.15e21T^{2} \)
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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

−10.02656455499075961404287939680, −9.154192287018676203948781320582, −8.317305535937282314860100280507, −6.87969885738030070856029577020, −5.75302197749767168771143427918, −4.79804065845696392993226577056, −3.97540415668200131972463613495, −2.44247104235441647077859346312, −1.82083742375174775200983468380, 0, 1.82083742375174775200983468380, 2.44247104235441647077859346312, 3.97540415668200131972463613495, 4.79804065845696392993226577056, 5.75302197749767168771143427918, 6.87969885738030070856029577020, 8.317305535937282314860100280507, 9.154192287018676203948781320582, 10.02656455499075961404287939680

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