Properties

Label 2-177-1.1-c9-0-85
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  + 35.2·2-s + 81·3-s + 733.·4-s + 388.·5-s + 2.85e3·6-s − 2.37e3·7-s + 7.80e3·8-s + 6.56e3·9-s + 1.37e4·10-s − 8.87e4·11-s + 5.93e4·12-s − 9.50e4·13-s − 8.38e4·14-s + 3.15e4·15-s − 9.99e4·16-s + 4.44e5·17-s + 2.31e5·18-s − 7.70e5·19-s + 2.85e5·20-s − 1.92e5·21-s − 3.13e6·22-s − 7.66e5·23-s + 6.32e5·24-s − 1.80e6·25-s − 3.35e6·26-s + 5.31e5·27-s − 1.74e6·28-s + ⋯
L(s)  = 1  + 1.55·2-s + 0.577·3-s + 1.43·4-s + 0.278·5-s + 0.900·6-s − 0.374·7-s + 0.673·8-s + 0.333·9-s + 0.433·10-s − 1.82·11-s + 0.826·12-s − 0.922·13-s − 0.583·14-s + 0.160·15-s − 0.381·16-s + 1.29·17-s + 0.519·18-s − 1.35·19-s + 0.398·20-s − 0.215·21-s − 2.85·22-s − 0.570·23-s + 0.389·24-s − 0.922·25-s − 1.43·26-s + 0.192·27-s − 0.535·28-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 \( 1 - 35.2T + 512T^{2} \)
5 \( 1 - 388.T + 1.95e6T^{2} \)
7 \( 1 + 2.37e3T + 4.03e7T^{2} \)
11 \( 1 + 8.87e4T + 2.35e9T^{2} \)
13 \( 1 + 9.50e4T + 1.06e10T^{2} \)
17 \( 1 - 4.44e5T + 1.18e11T^{2} \)
19 \( 1 + 7.70e5T + 3.22e11T^{2} \)
23 \( 1 + 7.66e5T + 1.80e12T^{2} \)
29 \( 1 - 3.07e6T + 1.45e13T^{2} \)
31 \( 1 + 4.59e6T + 2.64e13T^{2} \)
37 \( 1 - 8.69e6T + 1.29e14T^{2} \)
41 \( 1 - 3.69e6T + 3.27e14T^{2} \)
43 \( 1 + 1.75e7T + 5.02e14T^{2} \)
47 \( 1 - 3.58e7T + 1.11e15T^{2} \)
53 \( 1 - 7.15e7T + 3.29e15T^{2} \)
61 \( 1 - 1.49e8T + 1.16e16T^{2} \)
67 \( 1 + 1.05e8T + 2.72e16T^{2} \)
71 \( 1 + 4.67e7T + 4.58e16T^{2} \)
73 \( 1 + 4.65e7T + 5.88e16T^{2} \)
79 \( 1 - 2.64e8T + 1.19e17T^{2} \)
83 \( 1 + 8.34e8T + 1.86e17T^{2} \)
89 \( 1 - 2.19e8T + 3.50e17T^{2} \)
97 \( 1 + 6.96e8T + 7.60e17T^{2} \)
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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

−10.54609886037467471046567441057, −9.772966693534542669631730530243, −8.225913245465062541737020573834, −7.24329676101923717681977166596, −5.93308020020221013167451717579, −5.13319682030703501690053682359, −4.00619967621976293505121966001, −2.84025748571903734453087527659, −2.18585614504344973319607872995, 0, 2.18585614504344973319607872995, 2.84025748571903734453087527659, 4.00619967621976293505121966001, 5.13319682030703501690053682359, 5.93308020020221013167451717579, 7.24329676101923717681977166596, 8.225913245465062541737020573834, 9.772966693534542669631730530243, 10.54609886037467471046567441057

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