Properties

Label 2-177-1.1-c11-0-61
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 35.6·2-s − 243·3-s − 779.·4-s − 1.84e3·5-s + 8.65e3·6-s + 6.41e4·7-s + 1.00e5·8-s + 5.90e4·9-s + 6.58e4·10-s − 2.56e5·11-s + 1.89e5·12-s + 1.02e6·13-s − 2.28e6·14-s + 4.49e5·15-s − 1.98e6·16-s − 1.90e6·17-s − 2.10e6·18-s − 1.21e7·19-s + 1.44e6·20-s − 1.55e7·21-s + 9.13e6·22-s − 3.64e6·23-s − 2.44e7·24-s − 4.54e7·25-s − 3.65e7·26-s − 1.43e7·27-s − 5.00e7·28-s + ⋯
L(s)  = 1  − 0.786·2-s − 0.577·3-s − 0.380·4-s − 0.264·5-s + 0.454·6-s + 1.44·7-s + 1.08·8-s + 0.333·9-s + 0.208·10-s − 0.480·11-s + 0.219·12-s + 0.767·13-s − 1.13·14-s + 0.152·15-s − 0.474·16-s − 0.325·17-s − 0.262·18-s − 1.12·19-s + 0.100·20-s − 0.832·21-s + 0.378·22-s − 0.118·23-s − 0.627·24-s − 0.929·25-s − 0.604·26-s − 0.192·27-s − 0.549·28-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: \(1\)
Selberg data: \((2,\ 177,\ (\ :11/2),\ -1)\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) 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 \( 1 + 35.6T + 2.04e3T^{2} \)
5 \( 1 + 1.84e3T + 4.88e7T^{2} \)
7 \( 1 - 6.41e4T + 1.97e9T^{2} \)
11 \( 1 + 2.56e5T + 2.85e11T^{2} \)
13 \( 1 - 1.02e6T + 1.79e12T^{2} \)
17 \( 1 + 1.90e6T + 3.42e13T^{2} \)
19 \( 1 + 1.21e7T + 1.16e14T^{2} \)
23 \( 1 + 3.64e6T + 9.52e14T^{2} \)
29 \( 1 - 2.53e7T + 1.22e16T^{2} \)
31 \( 1 - 5.61e7T + 2.54e16T^{2} \)
37 \( 1 + 5.07e7T + 1.77e17T^{2} \)
41 \( 1 - 1.16e9T + 5.50e17T^{2} \)
43 \( 1 + 1.61e8T + 9.29e17T^{2} \)
47 \( 1 + 7.75e6T + 2.47e18T^{2} \)
53 \( 1 - 3.09e9T + 9.26e18T^{2} \)
61 \( 1 + 6.22e9T + 4.35e19T^{2} \)
67 \( 1 + 1.55e10T + 1.22e20T^{2} \)
71 \( 1 + 2.68e10T + 2.31e20T^{2} \)
73 \( 1 - 2.50e10T + 3.13e20T^{2} \)
79 \( 1 + 1.58e10T + 7.47e20T^{2} \)
83 \( 1 - 3.36e10T + 1.28e21T^{2} \)
89 \( 1 - 1.40e10T + 2.77e21T^{2} \)
97 \( 1 - 7.46e10T + 7.15e21T^{2} \)
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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

−10.36195087416771109967263314852, −8.994985097974120103199595558669, −8.198116512877497169953597224791, −7.48706671265016323817056463179, −5.98185258749755030212866504243, −4.79277097622927533252222961044, −4.09758243482247195308062766824, −2.03623826748461738775312881054, −1.05648912527521216764720239210, 0, 1.05648912527521216764720239210, 2.03623826748461738775312881054, 4.09758243482247195308062766824, 4.79277097622927533252222961044, 5.98185258749755030212866504243, 7.48706671265016323817056463179, 8.198116512877497169953597224791, 8.994985097974120103199595558669, 10.36195087416771109967263314852

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