Properties

Label 2-177-1.1-c7-0-17
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 14.7·2-s − 27·3-s + 91.0·4-s − 296.·5-s + 399.·6-s − 1.41e3·7-s + 547.·8-s + 729·9-s + 4.38e3·10-s − 7.53e3·11-s − 2.45e3·12-s + 7.06e3·13-s + 2.08e4·14-s + 8.00e3·15-s − 1.97e4·16-s + 1.48e4·17-s − 1.07e4·18-s − 2.40e4·19-s − 2.69e4·20-s + 3.80e4·21-s + 1.11e5·22-s + 4.15e4·23-s − 1.47e4·24-s + 9.80e3·25-s − 1.04e5·26-s − 1.96e4·27-s − 1.28e5·28-s + ⋯
L(s)  = 1  − 1.30·2-s − 0.577·3-s + 0.711·4-s − 1.06·5-s + 0.755·6-s − 1.55·7-s + 0.378·8-s + 0.333·9-s + 1.38·10-s − 1.70·11-s − 0.410·12-s + 0.892·13-s + 2.03·14-s + 0.612·15-s − 1.20·16-s + 0.732·17-s − 0.436·18-s − 0.803·19-s − 0.754·20-s + 0.897·21-s + 2.23·22-s + 0.712·23-s − 0.218·24-s + 0.125·25-s − 1.16·26-s − 0.192·27-s − 1.10·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 + 14.7T + 128T^{2} \)
5 \( 1 + 296.T + 7.81e4T^{2} \)
7 \( 1 + 1.41e3T + 8.23e5T^{2} \)
11 \( 1 + 7.53e3T + 1.94e7T^{2} \)
13 \( 1 - 7.06e3T + 6.27e7T^{2} \)
17 \( 1 - 1.48e4T + 4.10e8T^{2} \)
19 \( 1 + 2.40e4T + 8.93e8T^{2} \)
23 \( 1 - 4.15e4T + 3.40e9T^{2} \)
29 \( 1 - 1.57e4T + 1.72e10T^{2} \)
31 \( 1 - 1.08e5T + 2.75e10T^{2} \)
37 \( 1 - 3.31e5T + 9.49e10T^{2} \)
41 \( 1 - 2.24e5T + 1.94e11T^{2} \)
43 \( 1 + 1.34e5T + 2.71e11T^{2} \)
47 \( 1 - 8.16e5T + 5.06e11T^{2} \)
53 \( 1 - 1.21e6T + 1.17e12T^{2} \)
61 \( 1 - 1.88e6T + 3.14e12T^{2} \)
67 \( 1 + 2.09e6T + 6.06e12T^{2} \)
71 \( 1 + 1.80e6T + 9.09e12T^{2} \)
73 \( 1 + 4.93e6T + 1.10e13T^{2} \)
79 \( 1 + 8.26e6T + 1.92e13T^{2} \)
83 \( 1 + 2.95e6T + 2.71e13T^{2} \)
89 \( 1 - 3.82e6T + 4.42e13T^{2} \)
97 \( 1 + 6.24e6T + 8.07e13T^{2} \)
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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

−10.55961593722697086216410974682, −10.06415182247132923272336029678, −8.831985853887272903406410170785, −7.890268100049190470794606520326, −7.05448766698217224135325845910, −5.81780227344300270267591646152, −4.21180787431480539445966581307, −2.82248641663990520093085642815, −0.77668908563956621872924272170, 0, 0.77668908563956621872924272170, 2.82248641663990520093085642815, 4.21180787431480539445966581307, 5.81780227344300270267591646152, 7.05448766698217224135325845910, 7.890268100049190470794606520326, 8.831985853887272903406410170785, 10.06415182247132923272336029678, 10.55961593722697086216410974682

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