Properties

Label 2-177-1.1-c9-0-78
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  + 24.3·2-s + 81·3-s + 81.7·4-s + 1.96e3·5-s + 1.97e3·6-s − 1.14e4·7-s − 1.04e4·8-s + 6.56e3·9-s + 4.79e4·10-s − 163.·11-s + 6.61e3·12-s + 3.46e4·13-s − 2.78e5·14-s + 1.59e5·15-s − 2.97e5·16-s + 1.70e5·17-s + 1.59e5·18-s + 5.27e5·19-s + 1.60e5·20-s − 9.24e5·21-s − 3.98e3·22-s − 1.60e6·23-s − 8.49e5·24-s + 1.92e6·25-s + 8.44e5·26-s + 5.31e5·27-s − 9.32e5·28-s + ⋯
L(s)  = 1  + 1.07·2-s + 0.577·3-s + 0.159·4-s + 1.40·5-s + 0.621·6-s − 1.79·7-s − 0.904·8-s + 0.333·9-s + 1.51·10-s − 0.00336·11-s + 0.0921·12-s + 0.336·13-s − 1.93·14-s + 0.813·15-s − 1.13·16-s + 0.495·17-s + 0.358·18-s + 0.928·19-s + 0.224·20-s − 1.03·21-s − 0.00362·22-s − 1.19·23-s − 0.522·24-s + 0.986·25-s + 0.362·26-s + 0.192·27-s − 0.286·28-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 \( 1 - 24.3T + 512T^{2} \)
5 \( 1 - 1.96e3T + 1.95e6T^{2} \)
7 \( 1 + 1.14e4T + 4.03e7T^{2} \)
11 \( 1 + 163.T + 2.35e9T^{2} \)
13 \( 1 - 3.46e4T + 1.06e10T^{2} \)
17 \( 1 - 1.70e5T + 1.18e11T^{2} \)
19 \( 1 - 5.27e5T + 3.22e11T^{2} \)
23 \( 1 + 1.60e6T + 1.80e12T^{2} \)
29 \( 1 + 3.05e6T + 1.45e13T^{2} \)
31 \( 1 + 7.55e6T + 2.64e13T^{2} \)
37 \( 1 + 1.02e7T + 1.29e14T^{2} \)
41 \( 1 + 3.13e7T + 3.27e14T^{2} \)
43 \( 1 + 4.71e6T + 5.02e14T^{2} \)
47 \( 1 + 6.57e7T + 1.11e15T^{2} \)
53 \( 1 - 9.26e7T + 3.29e15T^{2} \)
61 \( 1 - 1.91e7T + 1.16e16T^{2} \)
67 \( 1 + 2.95e8T + 2.72e16T^{2} \)
71 \( 1 - 3.68e8T + 4.58e16T^{2} \)
73 \( 1 + 4.08e8T + 5.88e16T^{2} \)
79 \( 1 + 3.77e8T + 1.19e17T^{2} \)
83 \( 1 - 5.62e8T + 1.86e17T^{2} \)
89 \( 1 + 9.87e8T + 3.50e17T^{2} \)
97 \( 1 - 5.29e8T + 7.60e17T^{2} \)
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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

−10.11232093273336008633017958363, −9.648884645140850780419473629759, −8.806065182877428688813417116045, −6.98704831941114185120855619103, −6.03029502214827558685159767969, −5.38230025788717289789451134351, −3.68805334438910312948657643189, −3.09913614288342444299001913377, −1.84359489589368202249586098984, 0, 1.84359489589368202249586098984, 3.09913614288342444299001913377, 3.68805334438910312948657643189, 5.38230025788717289789451134351, 6.03029502214827558685159767969, 6.98704831941114185120855619103, 8.806065182877428688813417116045, 9.648884645140850780419473629759, 10.11232093273336008633017958363

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