Properties

Label 2-177-1.1-c13-0-26
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $189.798$
Root an. cond. $13.7767$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 116.·2-s + 729·3-s + 5.44e3·4-s − 2.32e4·5-s + 8.51e4·6-s − 5.57e5·7-s − 3.20e5·8-s + 5.31e5·9-s − 2.71e6·10-s + 2.68e6·11-s + 3.97e6·12-s − 2.49e7·13-s − 6.50e7·14-s − 1.69e7·15-s − 8.20e7·16-s + 1.26e8·17-s + 6.20e7·18-s − 2.65e8·19-s − 1.26e8·20-s − 4.06e8·21-s + 3.13e8·22-s + 2.77e8·23-s − 2.33e8·24-s − 6.78e8·25-s − 2.91e9·26-s + 3.87e8·27-s − 3.03e9·28-s + ⋯
L(s)  = 1  + 1.29·2-s + 0.577·3-s + 0.665·4-s − 0.666·5-s + 0.745·6-s − 1.78·7-s − 0.431·8-s + 0.333·9-s − 0.860·10-s + 0.457·11-s + 0.384·12-s − 1.43·13-s − 2.30·14-s − 0.384·15-s − 1.22·16-s + 1.27·17-s + 0.430·18-s − 1.29·19-s − 0.443·20-s − 1.03·21-s + 0.590·22-s + 0.390·23-s − 0.249·24-s − 0.555·25-s − 1.85·26-s + 0.192·27-s − 1.19·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(189.798\)
Root analytic conductor: \(13.7767\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :13/2),\ 1)\)

Particular Values

\(L(7)\) \(\approx\) \(2.313410517\)
\(L(\frac12)\) \(\approx\) \(2.313410517\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 729T \)
59 \( 1 + 4.21e10T \)
good2 \( 1 - 116.T + 8.19e3T^{2} \)
5 \( 1 + 2.32e4T + 1.22e9T^{2} \)
7 \( 1 + 5.57e5T + 9.68e10T^{2} \)
11 \( 1 - 2.68e6T + 3.45e13T^{2} \)
13 \( 1 + 2.49e7T + 3.02e14T^{2} \)
17 \( 1 - 1.26e8T + 9.90e15T^{2} \)
19 \( 1 + 2.65e8T + 4.20e16T^{2} \)
23 \( 1 - 2.77e8T + 5.04e17T^{2} \)
29 \( 1 - 4.45e9T + 1.02e19T^{2} \)
31 \( 1 - 5.87e9T + 2.44e19T^{2} \)
37 \( 1 + 7.00e9T + 2.43e20T^{2} \)
41 \( 1 + 8.11e9T + 9.25e20T^{2} \)
43 \( 1 + 3.05e10T + 1.71e21T^{2} \)
47 \( 1 + 3.22e10T + 5.46e21T^{2} \)
53 \( 1 + 4.59e10T + 2.60e22T^{2} \)
61 \( 1 - 5.48e11T + 1.61e23T^{2} \)
67 \( 1 - 7.48e11T + 5.48e23T^{2} \)
71 \( 1 + 1.36e11T + 1.16e24T^{2} \)
73 \( 1 + 4.36e11T + 1.67e24T^{2} \)
79 \( 1 - 3.74e12T + 4.66e24T^{2} \)
83 \( 1 - 2.29e12T + 8.87e24T^{2} \)
89 \( 1 + 3.64e12T + 2.19e25T^{2} \)
97 \( 1 - 1.41e13T + 6.73e25T^{2} \)
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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

−10.17161366699413521625425324305, −9.499479967598864250453486338543, −8.285827715220964406796282304219, −6.95158941113140591250023095207, −6.30255921940973311772206158375, −4.95265637473520522085895777461, −3.93075328045286281442790279575, −3.22831734200491475529846174737, −2.46658333106325163359842034496, −0.48123497636585766289356444024, 0.48123497636585766289356444024, 2.46658333106325163359842034496, 3.22831734200491475529846174737, 3.93075328045286281442790279575, 4.95265637473520522085895777461, 6.30255921940973311772206158375, 6.95158941113140591250023095207, 8.285827715220964406796282304219, 9.499479967598864250453486338543, 10.17161366699413521625425324305

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