Properties

Label 2-177-1.1-c11-0-7
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  + 45.0·2-s + 243·3-s − 17.2·4-s − 1.24e4·5-s + 1.09e4·6-s − 1.13e4·7-s − 9.30e4·8-s + 5.90e4·9-s − 5.60e5·10-s − 6.24e5·11-s − 4.18e3·12-s − 2.19e6·13-s − 5.10e5·14-s − 3.02e6·15-s − 4.15e6·16-s − 2.69e6·17-s + 2.66e6·18-s − 1.40e7·19-s + 2.14e5·20-s − 2.75e6·21-s − 2.81e7·22-s + 2.01e7·23-s − 2.26e7·24-s + 1.06e8·25-s − 9.89e7·26-s + 1.43e7·27-s + 1.94e5·28-s + ⋯
L(s)  = 1  + 0.995·2-s + 0.577·3-s − 0.00840·4-s − 1.78·5-s + 0.574·6-s − 0.254·7-s − 1.00·8-s + 0.333·9-s − 1.77·10-s − 1.16·11-s − 0.00485·12-s − 1.63·13-s − 0.253·14-s − 1.02·15-s − 0.991·16-s − 0.460·17-s + 0.331·18-s − 1.30·19-s + 0.0149·20-s − 0.147·21-s − 1.16·22-s + 0.652·23-s − 0.579·24-s + 2.17·25-s − 1.63·26-s + 0.192·27-s + 0.00213·28-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

\(L(6)\) \(\approx\) \(0.4080266419\)
\(L(\frac12)\) \(\approx\) \(0.4080266419\)
\(L(\frac{13}{2})\) 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 \( 1 - 45.0T + 2.04e3T^{2} \)
5 \( 1 + 1.24e4T + 4.88e7T^{2} \)
7 \( 1 + 1.13e4T + 1.97e9T^{2} \)
11 \( 1 + 6.24e5T + 2.85e11T^{2} \)
13 \( 1 + 2.19e6T + 1.79e12T^{2} \)
17 \( 1 + 2.69e6T + 3.42e13T^{2} \)
19 \( 1 + 1.40e7T + 1.16e14T^{2} \)
23 \( 1 - 2.01e7T + 9.52e14T^{2} \)
29 \( 1 + 3.17e6T + 1.22e16T^{2} \)
31 \( 1 - 1.00e8T + 2.54e16T^{2} \)
37 \( 1 - 7.58e8T + 1.77e17T^{2} \)
41 \( 1 + 7.54e6T + 5.50e17T^{2} \)
43 \( 1 + 1.42e9T + 9.29e17T^{2} \)
47 \( 1 + 1.00e9T + 2.47e18T^{2} \)
53 \( 1 + 3.23e9T + 9.26e18T^{2} \)
61 \( 1 + 9.23e9T + 4.35e19T^{2} \)
67 \( 1 + 1.91e9T + 1.22e20T^{2} \)
71 \( 1 + 1.76e10T + 2.31e20T^{2} \)
73 \( 1 - 1.51e10T + 3.13e20T^{2} \)
79 \( 1 - 7.55e9T + 7.47e20T^{2} \)
83 \( 1 - 5.47e10T + 1.28e21T^{2} \)
89 \( 1 + 3.81e10T + 2.77e21T^{2} \)
97 \( 1 + 5.92e10T + 7.15e21T^{2} \)
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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

−10.90936059299731880615857749449, −9.604042643274786082953711317646, −8.388788982350576341529821154086, −7.70779101422498791765303768200, −6.60325143517140948464405065908, −4.83690936335379919205844039256, −4.46210573685591752995312709595, −3.25777517226298835654325158194, −2.56053671377591200005945712577, −0.22395990058319586127536642344, 0.22395990058319586127536642344, 2.56053671377591200005945712577, 3.25777517226298835654325158194, 4.46210573685591752995312709595, 4.83690936335379919205844039256, 6.60325143517140948464405065908, 7.70779101422498791765303768200, 8.388788982350576341529821154086, 9.604042643274786082953711317646, 10.90936059299731880615857749449

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