Properties

Label 2-177-1.1-c9-0-76
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  + 6.97·2-s + 81·3-s − 463.·4-s + 1.65e3·5-s + 564.·6-s + 2.03e3·7-s − 6.80e3·8-s + 6.56e3·9-s + 1.15e4·10-s + 9.31e3·11-s − 3.75e4·12-s − 1.11e5·13-s + 1.42e4·14-s + 1.34e5·15-s + 1.89e5·16-s − 4.99e5·17-s + 4.57e4·18-s + 5.72e5·19-s − 7.67e5·20-s + 1.65e5·21-s + 6.49e4·22-s − 1.77e6·23-s − 5.50e5·24-s + 7.90e5·25-s − 7.76e5·26-s + 5.31e5·27-s − 9.44e5·28-s + ⋯
L(s)  = 1  + 0.308·2-s + 0.577·3-s − 0.905·4-s + 1.18·5-s + 0.177·6-s + 0.320·7-s − 0.587·8-s + 0.333·9-s + 0.365·10-s + 0.191·11-s − 0.522·12-s − 1.08·13-s + 0.0988·14-s + 0.684·15-s + 0.724·16-s − 1.45·17-s + 0.102·18-s + 1.00·19-s − 1.07·20-s + 0.185·21-s + 0.0591·22-s − 1.32·23-s − 0.338·24-s + 0.404·25-s − 0.333·26-s + 0.192·27-s − 0.290·28-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 \( 1 - 6.97T + 512T^{2} \)
5 \( 1 - 1.65e3T + 1.95e6T^{2} \)
7 \( 1 - 2.03e3T + 4.03e7T^{2} \)
11 \( 1 - 9.31e3T + 2.35e9T^{2} \)
13 \( 1 + 1.11e5T + 1.06e10T^{2} \)
17 \( 1 + 4.99e5T + 1.18e11T^{2} \)
19 \( 1 - 5.72e5T + 3.22e11T^{2} \)
23 \( 1 + 1.77e6T + 1.80e12T^{2} \)
29 \( 1 + 1.11e6T + 1.45e13T^{2} \)
31 \( 1 + 2.68e6T + 2.64e13T^{2} \)
37 \( 1 - 6.12e6T + 1.29e14T^{2} \)
41 \( 1 - 7.80e6T + 3.27e14T^{2} \)
43 \( 1 + 8.86e6T + 5.02e14T^{2} \)
47 \( 1 - 8.20e6T + 1.11e15T^{2} \)
53 \( 1 + 2.44e7T + 3.29e15T^{2} \)
61 \( 1 - 9.11e7T + 1.16e16T^{2} \)
67 \( 1 + 2.68e7T + 2.72e16T^{2} \)
71 \( 1 + 3.59e8T + 4.58e16T^{2} \)
73 \( 1 + 1.80e8T + 5.88e16T^{2} \)
79 \( 1 + 4.32e8T + 1.19e17T^{2} \)
83 \( 1 + 4.86e8T + 1.86e17T^{2} \)
89 \( 1 - 3.77e8T + 3.50e17T^{2} \)
97 \( 1 + 2.52e8T + 7.60e17T^{2} \)
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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

−10.10275840646508159916319864442, −9.524190816963143521481805008259, −8.718254231544706547342727568517, −7.51416419393818714873599763383, −6.11489775633363560446459830987, −5.08786749996210033668551113546, −4.11335264778647375633903361631, −2.66828251076541522505431583254, −1.62925308719520663546726775519, 0, 1.62925308719520663546726775519, 2.66828251076541522505431583254, 4.11335264778647375633903361631, 5.08786749996210033668551113546, 6.11489775633363560446459830987, 7.51416419393818714873599763383, 8.718254231544706547342727568517, 9.524190816963143521481805008259, 10.10275840646508159916319864442

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