Properties

Label 2-177-1.1-c5-0-38
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  + 5.27·2-s − 9·3-s − 4.20·4-s − 17.5·5-s − 47.4·6-s + 172.·7-s − 190.·8-s + 81·9-s − 92.4·10-s + 249.·11-s + 37.8·12-s + 282.·13-s + 910.·14-s + 157.·15-s − 871.·16-s − 1.34e3·17-s + 427.·18-s − 2.88e3·19-s + 73.7·20-s − 1.55e3·21-s + 1.31e3·22-s + 692.·23-s + 1.71e3·24-s − 2.81e3·25-s + 1.49e3·26-s − 729·27-s − 726.·28-s + ⋯
L(s)  = 1  + 0.931·2-s − 0.577·3-s − 0.131·4-s − 0.313·5-s − 0.538·6-s + 1.33·7-s − 1.05·8-s + 0.333·9-s − 0.292·10-s + 0.621·11-s + 0.0758·12-s + 0.464·13-s + 1.24·14-s + 0.181·15-s − 0.851·16-s − 1.13·17-s + 0.310·18-s − 1.83·19-s + 0.0412·20-s − 0.769·21-s + 0.579·22-s + 0.273·23-s + 0.608·24-s − 0.901·25-s + 0.432·26-s − 0.192·27-s − 0.175·28-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 \( 1 - 5.27T + 32T^{2} \)
5 \( 1 + 17.5T + 3.12e3T^{2} \)
7 \( 1 - 172.T + 1.68e4T^{2} \)
11 \( 1 - 249.T + 1.61e5T^{2} \)
13 \( 1 - 282.T + 3.71e5T^{2} \)
17 \( 1 + 1.34e3T + 1.41e6T^{2} \)
19 \( 1 + 2.88e3T + 2.47e6T^{2} \)
23 \( 1 - 692.T + 6.43e6T^{2} \)
29 \( 1 + 2.77e3T + 2.05e7T^{2} \)
31 \( 1 + 4.10e3T + 2.86e7T^{2} \)
37 \( 1 + 1.06e4T + 6.93e7T^{2} \)
41 \( 1 + 828.T + 1.15e8T^{2} \)
43 \( 1 - 8.99e3T + 1.47e8T^{2} \)
47 \( 1 + 4.34e3T + 2.29e8T^{2} \)
53 \( 1 + 1.79e3T + 4.18e8T^{2} \)
61 \( 1 + 2.80e4T + 8.44e8T^{2} \)
67 \( 1 - 3.09e4T + 1.35e9T^{2} \)
71 \( 1 - 3.94e3T + 1.80e9T^{2} \)
73 \( 1 - 1.77e4T + 2.07e9T^{2} \)
79 \( 1 + 5.70e4T + 3.07e9T^{2} \)
83 \( 1 - 1.75e4T + 3.93e9T^{2} \)
89 \( 1 + 5.36e4T + 5.58e9T^{2} \)
97 \( 1 + 1.15e5T + 8.58e9T^{2} \)
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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

−11.39637369240011798106982737023, −10.81474807539144533494645917105, −9.117404203595384869698217638736, −8.270498837026490629677493230863, −6.77093323610824169783531742558, −5.69685737904594283415334147497, −4.58267620405430404759562377617, −3.92979710507895581360082710614, −1.87190997829875665314687028645, 0, 1.87190997829875665314687028645, 3.92979710507895581360082710614, 4.58267620405430404759562377617, 5.69685737904594283415334147497, 6.77093323610824169783531742558, 8.270498837026490629677493230863, 9.117404203595384869698217638736, 10.81474807539144533494645917105, 11.39637369240011798106982737023

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