Properties

Label 2-177-1.1-c11-0-58
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  + 77.0·2-s − 243·3-s + 3.88e3·4-s + 2.37e3·5-s − 1.87e4·6-s + 4.26e4·7-s + 1.41e5·8-s + 5.90e4·9-s + 1.83e5·10-s − 7.32e5·11-s − 9.45e5·12-s + 2.06e6·13-s + 3.28e6·14-s − 5.77e5·15-s + 2.96e6·16-s + 3.22e6·17-s + 4.54e6·18-s + 7.79e6·19-s + 9.24e6·20-s − 1.03e7·21-s − 5.64e7·22-s + 1.03e7·23-s − 3.44e7·24-s − 4.31e7·25-s + 1.59e8·26-s − 1.43e7·27-s + 1.65e8·28-s + ⋯
L(s)  = 1  + 1.70·2-s − 0.577·3-s + 1.89·4-s + 0.340·5-s − 0.983·6-s + 0.959·7-s + 1.53·8-s + 0.333·9-s + 0.578·10-s − 1.37·11-s − 1.09·12-s + 1.54·13-s + 1.63·14-s − 0.196·15-s + 0.707·16-s + 0.550·17-s + 0.567·18-s + 0.722·19-s + 0.645·20-s − 0.554·21-s − 2.33·22-s + 0.334·23-s − 0.883·24-s − 0.884·25-s + 2.63·26-s − 0.192·27-s + 1.82·28-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

\(L(6)\) \(\approx\) \(7.122780924\)
\(L(\frac12)\) \(\approx\) \(7.122780924\)
\(L(\frac{13}{2})\) 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 \( 1 - 77.0T + 2.04e3T^{2} \)
5 \( 1 - 2.37e3T + 4.88e7T^{2} \)
7 \( 1 - 4.26e4T + 1.97e9T^{2} \)
11 \( 1 + 7.32e5T + 2.85e11T^{2} \)
13 \( 1 - 2.06e6T + 1.79e12T^{2} \)
17 \( 1 - 3.22e6T + 3.42e13T^{2} \)
19 \( 1 - 7.79e6T + 1.16e14T^{2} \)
23 \( 1 - 1.03e7T + 9.52e14T^{2} \)
29 \( 1 - 1.21e8T + 1.22e16T^{2} \)
31 \( 1 + 4.95e7T + 2.54e16T^{2} \)
37 \( 1 - 5.42e8T + 1.77e17T^{2} \)
41 \( 1 + 1.04e9T + 5.50e17T^{2} \)
43 \( 1 + 8.93e7T + 9.29e17T^{2} \)
47 \( 1 - 2.62e9T + 2.47e18T^{2} \)
53 \( 1 - 5.24e9T + 9.26e18T^{2} \)
61 \( 1 + 4.55e8T + 4.35e19T^{2} \)
67 \( 1 - 1.20e10T + 1.22e20T^{2} \)
71 \( 1 - 6.37e9T + 2.31e20T^{2} \)
73 \( 1 + 6.24e8T + 3.13e20T^{2} \)
79 \( 1 - 6.22e9T + 7.47e20T^{2} \)
83 \( 1 + 3.66e10T + 1.28e21T^{2} \)
89 \( 1 + 9.06e10T + 2.77e21T^{2} \)
97 \( 1 + 3.76e10T + 7.15e21T^{2} \)
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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

−11.09442012385384069468452414445, −10.18036575584051607648032336393, −8.362677608097222717000471744463, −7.25673220713130276493477265736, −5.95379921345935803767896615926, −5.43990606216173172301493215476, −4.55134716882322490120278267467, −3.40811428937728612322200726025, −2.21073262216761908719823907158, −1.01476840850406057034644552906, 1.01476840850406057034644552906, 2.21073262216761908719823907158, 3.40811428937728612322200726025, 4.55134716882322490120278267467, 5.43990606216173172301493215476, 5.95379921345935803767896615926, 7.25673220713130276493477265736, 8.362677608097222717000471744463, 10.18036575584051607648032336393, 11.09442012385384069468452414445

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