Properties

Label 2-177-1.1-c11-0-31
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 31.6·2-s + 243·3-s − 1.04e3·4-s − 7.92e3·5-s + 7.68e3·6-s + 5.53e4·7-s − 9.79e4·8-s + 5.90e4·9-s − 2.50e5·10-s + 7.49e5·11-s − 2.54e5·12-s − 8.55e5·13-s + 1.75e6·14-s − 1.92e6·15-s − 9.49e5·16-s − 8.42e6·17-s + 1.86e6·18-s − 2.35e5·19-s + 8.30e6·20-s + 1.34e7·21-s + 2.36e7·22-s + 1.10e7·23-s − 2.37e7·24-s + 1.39e7·25-s − 2.70e7·26-s + 1.43e7·27-s − 5.80e7·28-s + ⋯
L(s)  = 1  + 0.698·2-s + 0.577·3-s − 0.511·4-s − 1.13·5-s + 0.403·6-s + 1.24·7-s − 1.05·8-s + 0.333·9-s − 0.791·10-s + 1.40·11-s − 0.295·12-s − 0.638·13-s + 0.870·14-s − 0.654·15-s − 0.226·16-s − 1.43·17-s + 0.232·18-s − 0.0218·19-s + 0.580·20-s + 0.719·21-s + 0.980·22-s + 0.357·23-s − 0.609·24-s + 0.284·25-s − 0.446·26-s + 0.192·27-s − 0.637·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(2.769759382\)
\(L(\frac12)\) \(\approx\) \(2.769759382\)
\(L(\frac{13}{2})\) 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 \( 1 - 31.6T + 2.04e3T^{2} \)
5 \( 1 + 7.92e3T + 4.88e7T^{2} \)
7 \( 1 - 5.53e4T + 1.97e9T^{2} \)
11 \( 1 - 7.49e5T + 2.85e11T^{2} \)
13 \( 1 + 8.55e5T + 1.79e12T^{2} \)
17 \( 1 + 8.42e6T + 3.42e13T^{2} \)
19 \( 1 + 2.35e5T + 1.16e14T^{2} \)
23 \( 1 - 1.10e7T + 9.52e14T^{2} \)
29 \( 1 + 3.07e7T + 1.22e16T^{2} \)
31 \( 1 - 9.66e7T + 2.54e16T^{2} \)
37 \( 1 + 7.15e8T + 1.77e17T^{2} \)
41 \( 1 - 1.96e8T + 5.50e17T^{2} \)
43 \( 1 - 1.69e9T + 9.29e17T^{2} \)
47 \( 1 + 3.75e8T + 2.47e18T^{2} \)
53 \( 1 + 2.44e9T + 9.26e18T^{2} \)
61 \( 1 - 1.16e9T + 4.35e19T^{2} \)
67 \( 1 - 3.45e9T + 1.22e20T^{2} \)
71 \( 1 - 2.89e10T + 2.31e20T^{2} \)
73 \( 1 + 3.54e9T + 3.13e20T^{2} \)
79 \( 1 - 3.21e10T + 7.47e20T^{2} \)
83 \( 1 - 5.97e10T + 1.28e21T^{2} \)
89 \( 1 - 8.65e10T + 2.77e21T^{2} \)
97 \( 1 + 1.01e11T + 7.15e21T^{2} \)
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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

−10.99398491375554399671910160337, −9.325208000358727540380491190930, −8.651321641995834621344550203332, −7.74582716014402043975790993189, −6.60352347264077638542309414113, −4.95164823890063357582261278513, −4.28850106533574590481381248733, −3.55466895895539271588440344550, −2.08097997539639206064184629703, −0.66445915179312607369970364398, 0.66445915179312607369970364398, 2.08097997539639206064184629703, 3.55466895895539271588440344550, 4.28850106533574590481381248733, 4.95164823890063357582261278513, 6.60352347264077638542309414113, 7.74582716014402043975790993189, 8.651321641995834621344550203332, 9.325208000358727540380491190930, 10.99398491375554399671910160337

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