Properties

Label 2-177-1.1-c9-0-84
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 39.7·2-s + 81·3-s + 1.07e3·4-s − 1.73e3·5-s + 3.22e3·6-s − 3.88e3·7-s + 2.21e4·8-s + 6.56e3·9-s − 6.88e4·10-s + 5.68e4·11-s + 8.66e4·12-s − 1.09e5·13-s − 1.54e5·14-s − 1.40e5·15-s + 3.35e5·16-s − 6.53e5·17-s + 2.60e5·18-s − 1.75e5·19-s − 1.85e6·20-s − 3.14e5·21-s + 2.26e6·22-s − 1.42e6·23-s + 1.79e6·24-s + 1.04e6·25-s − 4.36e6·26-s + 5.31e5·27-s − 4.15e6·28-s + ⋯
L(s)  = 1  + 1.75·2-s + 0.577·3-s + 2.09·4-s − 1.23·5-s + 1.01·6-s − 0.611·7-s + 1.91·8-s + 0.333·9-s − 2.17·10-s + 1.17·11-s + 1.20·12-s − 1.06·13-s − 1.07·14-s − 0.715·15-s + 1.27·16-s − 1.89·17-s + 0.585·18-s − 0.309·19-s − 2.58·20-s − 0.353·21-s + 2.05·22-s − 1.06·23-s + 1.10·24-s + 0.534·25-s − 1.87·26-s + 0.192·27-s − 1.27·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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 \( 1 - 39.7T + 512T^{2} \)
5 \( 1 + 1.73e3T + 1.95e6T^{2} \)
7 \( 1 + 3.88e3T + 4.03e7T^{2} \)
11 \( 1 - 5.68e4T + 2.35e9T^{2} \)
13 \( 1 + 1.09e5T + 1.06e10T^{2} \)
17 \( 1 + 6.53e5T + 1.18e11T^{2} \)
19 \( 1 + 1.75e5T + 3.22e11T^{2} \)
23 \( 1 + 1.42e6T + 1.80e12T^{2} \)
29 \( 1 - 7.18e6T + 1.45e13T^{2} \)
31 \( 1 + 6.64e6T + 2.64e13T^{2} \)
37 \( 1 - 8.37e6T + 1.29e14T^{2} \)
41 \( 1 - 1.93e7T + 3.27e14T^{2} \)
43 \( 1 + 2.28e7T + 5.02e14T^{2} \)
47 \( 1 + 5.30e7T + 1.11e15T^{2} \)
53 \( 1 + 5.99e7T + 3.29e15T^{2} \)
61 \( 1 - 1.37e8T + 1.16e16T^{2} \)
67 \( 1 + 2.40e8T + 2.72e16T^{2} \)
71 \( 1 - 2.40e8T + 4.58e16T^{2} \)
73 \( 1 - 1.45e8T + 5.88e16T^{2} \)
79 \( 1 + 4.07e8T + 1.19e17T^{2} \)
83 \( 1 - 3.98e8T + 1.86e17T^{2} \)
89 \( 1 + 2.64e6T + 3.50e17T^{2} \)
97 \( 1 - 1.02e9T + 7.60e17T^{2} \)
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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

−11.06205568712215941971318410815, −9.541356896743079549455165749439, −8.221595392311813673516888338585, −6.97660867124570417629521762452, −6.39791647459469847907086429452, −4.63223373410960529915606151524, −4.10365915321777699805605431487, −3.16140777716975577044587829834, −2.06590637559572616597988737958, 0, 2.06590637559572616597988737958, 3.16140777716975577044587829834, 4.10365915321777699805605431487, 4.63223373410960529915606151524, 6.39791647459469847907086429452, 6.97660867124570417629521762452, 8.221595392311813673516888338585, 9.541356896743079549455165749439, 11.06205568712215941971318410815

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