Properties

Label 2-177-1.1-c5-0-45
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  + 6.91·2-s + 9·3-s + 15.7·4-s − 11.1·5-s + 62.2·6-s − 193.·7-s − 112.·8-s + 81·9-s − 76.9·10-s − 125.·11-s + 142.·12-s + 1.11e3·13-s − 1.33e3·14-s − 100.·15-s − 1.27e3·16-s − 1.94e3·17-s + 559.·18-s − 2.11e3·19-s − 175.·20-s − 1.73e3·21-s − 864.·22-s − 2.53e3·23-s − 1.00e3·24-s − 3.00e3·25-s + 7.71e3·26-s + 729·27-s − 3.05e3·28-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.577·3-s + 0.493·4-s − 0.199·5-s + 0.705·6-s − 1.49·7-s − 0.619·8-s + 0.333·9-s − 0.243·10-s − 0.311·11-s + 0.284·12-s + 1.83·13-s − 1.82·14-s − 0.114·15-s − 1.24·16-s − 1.63·17-s + 0.407·18-s − 1.34·19-s − 0.0982·20-s − 0.860·21-s − 0.380·22-s − 1.00·23-s − 0.357·24-s − 0.960·25-s + 2.23·26-s + 0.192·27-s − 0.735·28-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 \( 1 - 6.91T + 32T^{2} \)
5 \( 1 + 11.1T + 3.12e3T^{2} \)
7 \( 1 + 193.T + 1.68e4T^{2} \)
11 \( 1 + 125.T + 1.61e5T^{2} \)
13 \( 1 - 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + 1.94e3T + 1.41e6T^{2} \)
19 \( 1 + 2.11e3T + 2.47e6T^{2} \)
23 \( 1 + 2.53e3T + 6.43e6T^{2} \)
29 \( 1 - 4.53e3T + 2.05e7T^{2} \)
31 \( 1 + 902.T + 2.86e7T^{2} \)
37 \( 1 - 1.52e4T + 6.93e7T^{2} \)
41 \( 1 + 8.45e3T + 1.15e8T^{2} \)
43 \( 1 + 9.99e3T + 1.47e8T^{2} \)
47 \( 1 - 2.70e3T + 2.29e8T^{2} \)
53 \( 1 - 2.33e4T + 4.18e8T^{2} \)
61 \( 1 + 2.91e4T + 8.44e8T^{2} \)
67 \( 1 + 9.69e3T + 1.35e9T^{2} \)
71 \( 1 + 6.70e3T + 1.80e9T^{2} \)
73 \( 1 - 5.18e4T + 2.07e9T^{2} \)
79 \( 1 + 7.08e4T + 3.07e9T^{2} \)
83 \( 1 - 3.19e4T + 3.93e9T^{2} \)
89 \( 1 - 7.86e4T + 5.58e9T^{2} \)
97 \( 1 - 2.14e4T + 8.58e9T^{2} \)
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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

−11.57840830707846919362715449457, −10.39287310371657117447852616030, −9.154077283802484816751963979125, −8.352218611031379589932367938300, −6.52054855763515670630576108535, −6.12328169831198028712228551780, −4.29053251326394826218770699554, −3.62941653775026072006100413011, −2.43163010351496519199752879567, 0, 2.43163010351496519199752879567, 3.62941653775026072006100413011, 4.29053251326394826218770699554, 6.12328169831198028712228551780, 6.52054855763515670630576108535, 8.352218611031379589932367938300, 9.154077283802484816751963979125, 10.39287310371657117447852616030, 11.57840830707846919362715449457

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