Properties

Label 2-177-1.1-c9-0-48
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $91.1613$
Root an. cond. $9.54784$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 40.5·2-s + 81·3-s + 1.13e3·4-s − 281.·5-s + 3.28e3·6-s − 1.19e4·7-s + 2.52e4·8-s + 6.56e3·9-s − 1.14e4·10-s + 3.80e4·11-s + 9.18e4·12-s + 1.24e5·13-s − 4.84e5·14-s − 2.28e4·15-s + 4.42e5·16-s + 4.83e5·17-s + 2.66e5·18-s + 1.18e5·19-s − 3.19e5·20-s − 9.67e5·21-s + 1.54e6·22-s + 1.81e6·23-s + 2.04e6·24-s − 1.87e6·25-s + 5.05e6·26-s + 5.31e5·27-s − 1.35e7·28-s + ⋯
L(s)  = 1  + 1.79·2-s + 0.577·3-s + 2.21·4-s − 0.201·5-s + 1.03·6-s − 1.88·7-s + 2.17·8-s + 0.333·9-s − 0.361·10-s + 0.783·11-s + 1.27·12-s + 1.20·13-s − 3.37·14-s − 0.116·15-s + 1.68·16-s + 1.40·17-s + 0.597·18-s + 0.209·19-s − 0.446·20-s − 1.08·21-s + 1.40·22-s + 1.34·23-s + 1.25·24-s − 0.959·25-s + 2.16·26-s + 0.192·27-s − 4.16·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(91.1613\)
Root analytic conductor: \(9.54784\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(8.251927318\)
\(L(\frac12)\) \(\approx\) \(8.251927318\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 81T \)
59 \( 1 + 1.21e7T \)
good2 \( 1 - 40.5T + 512T^{2} \)
5 \( 1 + 281.T + 1.95e6T^{2} \)
7 \( 1 + 1.19e4T + 4.03e7T^{2} \)
11 \( 1 - 3.80e4T + 2.35e9T^{2} \)
13 \( 1 - 1.24e5T + 1.06e10T^{2} \)
17 \( 1 - 4.83e5T + 1.18e11T^{2} \)
19 \( 1 - 1.18e5T + 3.22e11T^{2} \)
23 \( 1 - 1.81e6T + 1.80e12T^{2} \)
29 \( 1 - 1.59e6T + 1.45e13T^{2} \)
31 \( 1 - 3.75e6T + 2.64e13T^{2} \)
37 \( 1 - 1.86e7T + 1.29e14T^{2} \)
41 \( 1 - 2.23e5T + 3.27e14T^{2} \)
43 \( 1 - 9.66e6T + 5.02e14T^{2} \)
47 \( 1 + 2.98e6T + 1.11e15T^{2} \)
53 \( 1 + 3.40e7T + 3.29e15T^{2} \)
61 \( 1 + 1.41e8T + 1.16e16T^{2} \)
67 \( 1 - 2.24e7T + 2.72e16T^{2} \)
71 \( 1 + 2.18e8T + 4.58e16T^{2} \)
73 \( 1 + 2.38e8T + 5.88e16T^{2} \)
79 \( 1 + 4.97e7T + 1.19e17T^{2} \)
83 \( 1 + 5.31e8T + 1.86e17T^{2} \)
89 \( 1 + 1.52e6T + 3.50e17T^{2} \)
97 \( 1 - 1.14e9T + 7.60e17T^{2} \)
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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

−11.44738011647106546390022825528, −10.14102066397760599782350877173, −9.109913332848081315030864651980, −7.50603349236435317265142335267, −6.43609020316761290556106025479, −5.86998600940315692617681686384, −4.25233656858561225890602857030, −3.38365616294759084151228550882, −2.90682104372935137941585478532, −1.13235026194029287114534943108, 1.13235026194029287114534943108, 2.90682104372935137941585478532, 3.38365616294759084151228550882, 4.25233656858561225890602857030, 5.86998600940315692617681686384, 6.43609020316761290556106025479, 7.50603349236435317265142335267, 9.109913332848081315030864651980, 10.14102066397760599782350877173, 11.44738011647106546390022825528

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