Properties

Label 2-177-1.1-c7-0-44
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  − 13.3·2-s + 27·3-s + 49.7·4-s + 152.·5-s − 360.·6-s − 1.32e3·7-s + 1.04e3·8-s + 729·9-s − 2.02e3·10-s + 5.13e3·11-s + 1.34e3·12-s − 1.22e4·13-s + 1.77e4·14-s + 4.10e3·15-s − 2.02e4·16-s + 2.66e4·17-s − 9.72e3·18-s + 3.31e4·19-s + 7.57e3·20-s − 3.58e4·21-s − 6.84e4·22-s − 9.70e4·23-s + 2.81e4·24-s − 5.49e4·25-s + 1.63e5·26-s + 1.96e4·27-s − 6.61e4·28-s + ⋯
L(s)  = 1  − 1.17·2-s + 0.577·3-s + 0.388·4-s + 0.544·5-s − 0.680·6-s − 1.46·7-s + 0.720·8-s + 0.333·9-s − 0.641·10-s + 1.16·11-s + 0.224·12-s − 1.54·13-s + 1.72·14-s + 0.314·15-s − 1.23·16-s + 1.31·17-s − 0.392·18-s + 1.11·19-s + 0.211·20-s − 0.845·21-s − 1.37·22-s − 1.66·23-s + 0.415·24-s − 0.703·25-s + 1.82·26-s + 0.192·27-s − 0.569·28-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 \( 1 + 13.3T + 128T^{2} \)
5 \( 1 - 152.T + 7.81e4T^{2} \)
7 \( 1 + 1.32e3T + 8.23e5T^{2} \)
11 \( 1 - 5.13e3T + 1.94e7T^{2} \)
13 \( 1 + 1.22e4T + 6.27e7T^{2} \)
17 \( 1 - 2.66e4T + 4.10e8T^{2} \)
19 \( 1 - 3.31e4T + 8.93e8T^{2} \)
23 \( 1 + 9.70e4T + 3.40e9T^{2} \)
29 \( 1 - 2.09e5T + 1.72e10T^{2} \)
31 \( 1 + 3.08e5T + 2.75e10T^{2} \)
37 \( 1 - 2.87e5T + 9.49e10T^{2} \)
41 \( 1 - 5.81e5T + 1.94e11T^{2} \)
43 \( 1 - 5.40e5T + 2.71e11T^{2} \)
47 \( 1 + 6.50e5T + 5.06e11T^{2} \)
53 \( 1 + 4.99e4T + 1.17e12T^{2} \)
61 \( 1 + 2.00e4T + 3.14e12T^{2} \)
67 \( 1 + 1.38e6T + 6.06e12T^{2} \)
71 \( 1 - 1.24e6T + 9.09e12T^{2} \)
73 \( 1 + 4.83e6T + 1.10e13T^{2} \)
79 \( 1 + 2.94e6T + 1.92e13T^{2} \)
83 \( 1 + 8.62e6T + 2.71e13T^{2} \)
89 \( 1 + 8.79e5T + 4.42e13T^{2} \)
97 \( 1 + 1.62e7T + 8.07e13T^{2} \)
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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

−10.17251015852899453151107777561, −9.608401673410879892210733256427, −9.357011556726651384110828800857, −7.88116001252629383530939410355, −7.08653566484324779476488342796, −5.82875043538230854705460521676, −4.04981811369366164148825535771, −2.70211298916659415480997093302, −1.33273985178876221463747688938, 0, 1.33273985178876221463747688938, 2.70211298916659415480997093302, 4.04981811369366164148825535771, 5.82875043538230854705460521676, 7.08653566484324779476488342796, 7.88116001252629383530939410355, 9.357011556726651384110828800857, 9.608401673410879892210733256427, 10.17251015852899453151107777561

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