Properties

Label 2-177-1.1-c9-0-67
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  + 35.4·2-s + 81·3-s + 745.·4-s + 1.62e3·5-s + 2.87e3·6-s + 5.25e3·7-s + 8.29e3·8-s + 6.56e3·9-s + 5.75e4·10-s + 6.32e4·11-s + 6.04e4·12-s − 9.31e4·13-s + 1.86e5·14-s + 1.31e5·15-s − 8.76e4·16-s + 3.12e5·17-s + 2.32e5·18-s + 2.95e5·19-s + 1.21e6·20-s + 4.25e5·21-s + 2.24e6·22-s − 6.60e5·23-s + 6.71e5·24-s + 6.80e5·25-s − 3.30e6·26-s + 5.31e5·27-s + 3.91e6·28-s + ⋯
L(s)  = 1  + 1.56·2-s + 0.577·3-s + 1.45·4-s + 1.16·5-s + 0.904·6-s + 0.827·7-s + 0.716·8-s + 0.333·9-s + 1.82·10-s + 1.30·11-s + 0.841·12-s − 0.905·13-s + 1.29·14-s + 0.670·15-s − 0.334·16-s + 0.907·17-s + 0.522·18-s + 0.519·19-s + 1.69·20-s + 0.477·21-s + 2.04·22-s − 0.492·23-s + 0.413·24-s + 0.348·25-s − 1.41·26-s + 0.192·27-s + 1.20·28-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

\(L(5)\) \(\approx\) \(9.997080929\)
\(L(\frac12)\) \(\approx\) \(9.997080929\)
\(L(\frac{11}{2})\) 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 \( 1 - 35.4T + 512T^{2} \)
5 \( 1 - 1.62e3T + 1.95e6T^{2} \)
7 \( 1 - 5.25e3T + 4.03e7T^{2} \)
11 \( 1 - 6.32e4T + 2.35e9T^{2} \)
13 \( 1 + 9.31e4T + 1.06e10T^{2} \)
17 \( 1 - 3.12e5T + 1.18e11T^{2} \)
19 \( 1 - 2.95e5T + 3.22e11T^{2} \)
23 \( 1 + 6.60e5T + 1.80e12T^{2} \)
29 \( 1 + 1.79e6T + 1.45e13T^{2} \)
31 \( 1 - 9.00e5T + 2.64e13T^{2} \)
37 \( 1 - 4.38e6T + 1.29e14T^{2} \)
41 \( 1 - 1.15e7T + 3.27e14T^{2} \)
43 \( 1 + 6.76e6T + 5.02e14T^{2} \)
47 \( 1 + 3.09e7T + 1.11e15T^{2} \)
53 \( 1 - 1.17e7T + 3.29e15T^{2} \)
61 \( 1 + 3.57e7T + 1.16e16T^{2} \)
67 \( 1 - 2.25e8T + 2.72e16T^{2} \)
71 \( 1 + 1.64e8T + 4.58e16T^{2} \)
73 \( 1 + 2.45e8T + 5.88e16T^{2} \)
79 \( 1 - 2.30e8T + 1.19e17T^{2} \)
83 \( 1 - 3.71e8T + 1.86e17T^{2} \)
89 \( 1 - 2.06e8T + 3.50e17T^{2} \)
97 \( 1 - 8.17e8T + 7.60e17T^{2} \)
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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

−11.44751086580250066260286858363, −9.976256076006101034034690057602, −9.170346475032040593123994475320, −7.69148775219944316795736473970, −6.49942434580435086990750770643, −5.54457501525427676069785670276, −4.62364450450684884619016573674, −3.50685058633982681325880179048, −2.31524378956277574853001653100, −1.43458142729025750442753680459, 1.43458142729025750442753680459, 2.31524378956277574853001653100, 3.50685058633982681325880179048, 4.62364450450684884619016573674, 5.54457501525427676069785670276, 6.49942434580435086990750770643, 7.69148775219944316795736473970, 9.170346475032040593123994475320, 9.976256076006101034034690057602, 11.44751086580250066260286858363

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