Properties

Label 2-177-1.1-c9-0-1
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  − 27.1·2-s − 81·3-s + 226.·4-s − 1.81e3·5-s + 2.20e3·6-s − 9.88e3·7-s + 7.75e3·8-s + 6.56e3·9-s + 4.92e4·10-s + 6.29e4·11-s − 1.83e4·12-s + 8.96e4·13-s + 2.68e5·14-s + 1.46e5·15-s − 3.26e5·16-s − 9.81e4·17-s − 1.78e5·18-s − 6.08e5·19-s − 4.10e5·20-s + 8.00e5·21-s − 1.71e6·22-s − 1.34e6·23-s − 6.28e5·24-s + 1.33e6·25-s − 2.43e6·26-s − 5.31e5·27-s − 2.23e6·28-s + ⋯
L(s)  = 1  − 1.20·2-s − 0.577·3-s + 0.442·4-s − 1.29·5-s + 0.693·6-s − 1.55·7-s + 0.669·8-s + 0.333·9-s + 1.55·10-s + 1.29·11-s − 0.255·12-s + 0.870·13-s + 1.86·14-s + 0.748·15-s − 1.24·16-s − 0.285·17-s − 0.400·18-s − 1.07·19-s − 0.573·20-s + 0.898·21-s − 1.55·22-s − 1.00·23-s − 0.386·24-s + 0.681·25-s − 1.04·26-s − 0.192·27-s − 0.688·28-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

\(L(5)\) \(\approx\) \(0.04442201481\)
\(L(\frac12)\) \(\approx\) \(0.04442201481\)
\(L(\frac{11}{2})\) 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 \( 1 + 27.1T + 512T^{2} \)
5 \( 1 + 1.81e3T + 1.95e6T^{2} \)
7 \( 1 + 9.88e3T + 4.03e7T^{2} \)
11 \( 1 - 6.29e4T + 2.35e9T^{2} \)
13 \( 1 - 8.96e4T + 1.06e10T^{2} \)
17 \( 1 + 9.81e4T + 1.18e11T^{2} \)
19 \( 1 + 6.08e5T + 3.22e11T^{2} \)
23 \( 1 + 1.34e6T + 1.80e12T^{2} \)
29 \( 1 - 1.65e6T + 1.45e13T^{2} \)
31 \( 1 + 3.82e6T + 2.64e13T^{2} \)
37 \( 1 + 2.43e5T + 1.29e14T^{2} \)
41 \( 1 + 3.05e7T + 3.27e14T^{2} \)
43 \( 1 + 9.67e6T + 5.02e14T^{2} \)
47 \( 1 + 1.94e7T + 1.11e15T^{2} \)
53 \( 1 + 6.14e7T + 3.29e15T^{2} \)
61 \( 1 + 1.88e8T + 1.16e16T^{2} \)
67 \( 1 - 7.34e7T + 2.72e16T^{2} \)
71 \( 1 + 1.30e8T + 4.58e16T^{2} \)
73 \( 1 - 3.50e7T + 5.88e16T^{2} \)
79 \( 1 + 1.62e8T + 1.19e17T^{2} \)
83 \( 1 - 1.68e8T + 1.86e17T^{2} \)
89 \( 1 + 9.09e8T + 3.50e17T^{2} \)
97 \( 1 - 2.90e7T + 7.60e17T^{2} \)
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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

−10.88008967422863461203634201524, −9.929968966092691650484728555024, −9.002266757051059542507143283093, −8.168682979610144813581337392080, −6.91065475072410366427124093269, −6.30578151451723988840483739693, −4.29793495965231152195578054391, −3.53988198978244068225714310158, −1.47654749076262994808241357424, −0.13522423139633208937914517139, 0.13522423139633208937914517139, 1.47654749076262994808241357424, 3.53988198978244068225714310158, 4.29793495965231152195578054391, 6.30578151451723988840483739693, 6.91065475072410366427124093269, 8.168682979610144813581337392080, 9.002266757051059542507143283093, 9.929968966092691650484728555024, 10.88008967422863461203634201524

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