Properties

Label 2-177-1.1-c7-0-62
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 13.6·2-s + 27·3-s + 58.8·4-s − 207.·5-s + 369.·6-s + 882.·7-s − 945.·8-s + 729·9-s − 2.83e3·10-s − 3.63e3·11-s + 1.58e3·12-s + 208.·13-s + 1.20e4·14-s − 5.59e3·15-s − 2.04e4·16-s − 1.31e4·17-s + 9.96e3·18-s − 9.91e3·19-s − 1.21e4·20-s + 2.38e4·21-s − 4.97e4·22-s − 3.77e4·23-s − 2.55e4·24-s − 3.51e4·25-s + 2.84e3·26-s + 1.96e4·27-s + 5.19e4·28-s + ⋯
L(s)  = 1  + 1.20·2-s + 0.577·3-s + 0.459·4-s − 0.742·5-s + 0.697·6-s + 0.972·7-s − 0.652·8-s + 0.333·9-s − 0.896·10-s − 0.823·11-s + 0.265·12-s + 0.0262·13-s + 1.17·14-s − 0.428·15-s − 1.24·16-s − 0.647·17-s + 0.402·18-s − 0.331·19-s − 0.340·20-s + 0.561·21-s − 0.995·22-s − 0.647·23-s − 0.376·24-s − 0.449·25-s + 0.0317·26-s + 0.192·27-s + 0.446·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 + 2.05e5T \)
good2 \( 1 - 13.6T + 128T^{2} \)
5 \( 1 + 207.T + 7.81e4T^{2} \)
7 \( 1 - 882.T + 8.23e5T^{2} \)
11 \( 1 + 3.63e3T + 1.94e7T^{2} \)
13 \( 1 - 208.T + 6.27e7T^{2} \)
17 \( 1 + 1.31e4T + 4.10e8T^{2} \)
19 \( 1 + 9.91e3T + 8.93e8T^{2} \)
23 \( 1 + 3.77e4T + 3.40e9T^{2} \)
29 \( 1 + 9.80e4T + 1.72e10T^{2} \)
31 \( 1 - 3.26e5T + 2.75e10T^{2} \)
37 \( 1 - 3.94e4T + 9.49e10T^{2} \)
41 \( 1 + 7.19e5T + 1.94e11T^{2} \)
43 \( 1 + 8.78e5T + 2.71e11T^{2} \)
47 \( 1 + 1.58e5T + 5.06e11T^{2} \)
53 \( 1 + 1.83e6T + 1.17e12T^{2} \)
61 \( 1 - 3.49e6T + 3.14e12T^{2} \)
67 \( 1 + 1.00e6T + 6.06e12T^{2} \)
71 \( 1 - 3.31e5T + 9.09e12T^{2} \)
73 \( 1 + 5.27e5T + 1.10e13T^{2} \)
79 \( 1 + 1.50e6T + 1.92e13T^{2} \)
83 \( 1 - 2.48e6T + 2.71e13T^{2} \)
89 \( 1 - 6.37e6T + 4.42e13T^{2} \)
97 \( 1 + 4.33e6T + 8.07e13T^{2} \)
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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

−11.32531427598809806580018312687, −9.988415783100917500615872235594, −8.511230921170854774840715938618, −7.925285517770011444566988202930, −6.52981879634007118352922136571, −5.09627325475728454898359087071, −4.35787023269503560951350897704, −3.28831013147514192232310597333, −2.02944580806888173636131037103, 0, 2.02944580806888173636131037103, 3.28831013147514192232310597333, 4.35787023269503560951350897704, 5.09627325475728454898359087071, 6.52981879634007118352922136571, 7.925285517770011444566988202930, 8.511230921170854774840715938618, 9.988415783100917500615872235594, 11.32531427598809806580018312687

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