Properties

Label 2-177-1.1-c5-0-35
Degree $2$
Conductor $177$
Sign $-1$
Analytic cond. $28.3879$
Root an. cond. $5.32803$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.21·2-s + 9·3-s − 30.5·4-s + 0.914·5-s − 10.9·6-s + 61.7·7-s + 76.0·8-s + 81·9-s − 1.11·10-s − 311.·11-s − 274.·12-s − 150.·13-s − 75.1·14-s + 8.23·15-s + 884.·16-s + 1.23e3·17-s − 98.5·18-s − 266.·19-s − 27.9·20-s + 555.·21-s + 378.·22-s − 1.92e3·23-s + 684.·24-s − 3.12e3·25-s + 182.·26-s + 729·27-s − 1.88e3·28-s + ⋯
L(s)  = 1  − 0.215·2-s + 0.577·3-s − 0.953·4-s + 0.0163·5-s − 0.124·6-s + 0.476·7-s + 0.420·8-s + 0.333·9-s − 0.00351·10-s − 0.776·11-s − 0.550·12-s − 0.246·13-s − 0.102·14-s + 0.00944·15-s + 0.863·16-s + 1.03·17-s − 0.0716·18-s − 0.169·19-s − 0.0156·20-s + 0.275·21-s + 0.166·22-s − 0.756·23-s + 0.242·24-s − 0.999·25-s + 0.0530·26-s + 0.192·27-s − 0.454·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-1$
Analytic conductor: \(28.3879\)
Root analytic conductor: \(5.32803\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 9T \)
59 \( 1 - 3.48e3T \)
good2 \( 1 + 1.21T + 32T^{2} \)
5 \( 1 - 0.914T + 3.12e3T^{2} \)
7 \( 1 - 61.7T + 1.68e4T^{2} \)
11 \( 1 + 311.T + 1.61e5T^{2} \)
13 \( 1 + 150.T + 3.71e5T^{2} \)
17 \( 1 - 1.23e3T + 1.41e6T^{2} \)
19 \( 1 + 266.T + 2.47e6T^{2} \)
23 \( 1 + 1.92e3T + 6.43e6T^{2} \)
29 \( 1 + 2.98e3T + 2.05e7T^{2} \)
31 \( 1 + 8.35e3T + 2.86e7T^{2} \)
37 \( 1 - 8.50e3T + 6.93e7T^{2} \)
41 \( 1 - 1.39e4T + 1.15e8T^{2} \)
43 \( 1 + 1.12e4T + 1.47e8T^{2} \)
47 \( 1 + 2.48e4T + 2.29e8T^{2} \)
53 \( 1 + 3.53e4T + 4.18e8T^{2} \)
61 \( 1 + 4.68e4T + 8.44e8T^{2} \)
67 \( 1 - 3.76e4T + 1.35e9T^{2} \)
71 \( 1 + 5.64e4T + 1.80e9T^{2} \)
73 \( 1 - 6.62e4T + 2.07e9T^{2} \)
79 \( 1 + 5.15e4T + 3.07e9T^{2} \)
83 \( 1 + 2.88e4T + 3.93e9T^{2} \)
89 \( 1 + 9.51e3T + 5.58e9T^{2} \)
97 \( 1 - 9.63e4T + 8.58e9T^{2} \)
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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

−11.18216089591956870585904961600, −9.990519371806501783544803430321, −9.361037089490606555391425866479, −8.066900405628620651623551043060, −7.69058322791114410644009812493, −5.74481078603826255624843191723, −4.63971921250433928994951929363, −3.41499865368432585025667328071, −1.72548427306702597091183689824, 0, 1.72548427306702597091183689824, 3.41499865368432585025667328071, 4.63971921250433928994951929363, 5.74481078603826255624843191723, 7.69058322791114410644009812493, 8.066900405628620651623551043060, 9.361037089490606555391425866479, 9.990519371806501783544803430321, 11.18216089591956870585904961600

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