Properties

Label 2-177-1.1-c11-0-6
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $135.996$
Root an. cond. $11.6617$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.21·2-s + 243·3-s − 2.02e3·4-s − 8.55e3·5-s − 1.26e3·6-s − 1.65e4·7-s + 2.12e4·8-s + 5.90e4·9-s + 4.45e4·10-s − 8.92e4·11-s − 4.91e5·12-s − 1.14e5·13-s + 8.61e4·14-s − 2.07e6·15-s + 4.02e6·16-s − 2.27e6·17-s − 3.07e5·18-s − 1.31e7·19-s + 1.72e7·20-s − 4.01e6·21-s + 4.65e5·22-s − 2.62e7·23-s + 5.15e6·24-s + 2.42e7·25-s + 5.98e5·26-s + 1.43e7·27-s + 3.33e7·28-s + ⋯
L(s)  = 1  − 0.115·2-s + 0.577·3-s − 0.986·4-s − 1.22·5-s − 0.0664·6-s − 0.371·7-s + 0.228·8-s + 0.333·9-s + 0.140·10-s − 0.167·11-s − 0.569·12-s − 0.0857·13-s + 0.0427·14-s − 0.706·15-s + 0.960·16-s − 0.388·17-s − 0.0383·18-s − 1.22·19-s + 1.20·20-s − 0.214·21-s + 0.0192·22-s − 0.851·23-s + 0.132·24-s + 0.497·25-s + 0.00987·26-s + 0.192·27-s + 0.366·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(135.996\)
Root analytic conductor: \(11.6617\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ 1)\)

Particular Values

\(L(6)\) \(\approx\) \(0.2283791978\)
\(L(\frac12)\) \(\approx\) \(0.2283791978\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 243T \)
59 \( 1 - 7.14e8T \)
good2 \( 1 + 5.21T + 2.04e3T^{2} \)
5 \( 1 + 8.55e3T + 4.88e7T^{2} \)
7 \( 1 + 1.65e4T + 1.97e9T^{2} \)
11 \( 1 + 8.92e4T + 2.85e11T^{2} \)
13 \( 1 + 1.14e5T + 1.79e12T^{2} \)
17 \( 1 + 2.27e6T + 3.42e13T^{2} \)
19 \( 1 + 1.31e7T + 1.16e14T^{2} \)
23 \( 1 + 2.62e7T + 9.52e14T^{2} \)
29 \( 1 + 1.02e8T + 1.22e16T^{2} \)
31 \( 1 + 2.81e8T + 2.54e16T^{2} \)
37 \( 1 + 9.42e7T + 1.77e17T^{2} \)
41 \( 1 + 1.20e9T + 5.50e17T^{2} \)
43 \( 1 + 6.09e8T + 9.29e17T^{2} \)
47 \( 1 + 2.59e9T + 2.47e18T^{2} \)
53 \( 1 - 5.94e9T + 9.26e18T^{2} \)
61 \( 1 - 4.54e9T + 4.35e19T^{2} \)
67 \( 1 - 9.68e9T + 1.22e20T^{2} \)
71 \( 1 - 1.70e10T + 2.31e20T^{2} \)
73 \( 1 + 8.07e9T + 3.13e20T^{2} \)
79 \( 1 - 3.36e10T + 7.47e20T^{2} \)
83 \( 1 + 2.87e10T + 1.28e21T^{2} \)
89 \( 1 + 2.72e9T + 2.77e21T^{2} \)
97 \( 1 - 1.22e11T + 7.15e21T^{2} \)
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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

−10.51790105292250625577626427648, −9.506084690152816564753051926165, −8.527048687078293915510730050977, −7.949597092538637886177135320704, −6.81090064117247315343238183129, −5.20486486276053989094035580127, −4.01686775132586466453326184273, −3.53348936756059100472626588248, −1.90072246352794431046007105346, −0.20783137167557155514127006537, 0.20783137167557155514127006537, 1.90072246352794431046007105346, 3.53348936756059100472626588248, 4.01686775132586466453326184273, 5.20486486276053989094035580127, 6.81090064117247315343238183129, 7.949597092538637886177135320704, 8.527048687078293915510730050977, 9.506084690152816564753051926165, 10.51790105292250625577626427648

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