Properties

Label 2-177-1.1-c5-0-46
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  + 10.7·2-s − 9·3-s + 83.9·4-s − 60.5·5-s − 96.8·6-s − 233.·7-s + 558.·8-s + 81·9-s − 652.·10-s + 441.·11-s − 755.·12-s − 615.·13-s − 2.51e3·14-s + 545.·15-s + 3.33e3·16-s − 1.49e3·17-s + 872.·18-s − 1.18e3·19-s − 5.08e3·20-s + 2.09e3·21-s + 4.75e3·22-s − 4.79e3·23-s − 5.02e3·24-s + 546.·25-s − 6.62e3·26-s − 729·27-s − 1.95e4·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.577·3-s + 2.62·4-s − 1.08·5-s − 1.09·6-s − 1.79·7-s + 3.08·8-s + 0.333·9-s − 2.06·10-s + 1.10·11-s − 1.51·12-s − 1.01·13-s − 3.42·14-s + 0.625·15-s + 3.25·16-s − 1.25·17-s + 0.634·18-s − 0.755·19-s − 2.84·20-s + 1.03·21-s + 2.09·22-s − 1.88·23-s − 1.78·24-s + 0.175·25-s − 1.92·26-s − 0.192·27-s − 4.71·28-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 \( 1 - 10.7T + 32T^{2} \)
5 \( 1 + 60.5T + 3.12e3T^{2} \)
7 \( 1 + 233.T + 1.68e4T^{2} \)
11 \( 1 - 441.T + 1.61e5T^{2} \)
13 \( 1 + 615.T + 3.71e5T^{2} \)
17 \( 1 + 1.49e3T + 1.41e6T^{2} \)
19 \( 1 + 1.18e3T + 2.47e6T^{2} \)
23 \( 1 + 4.79e3T + 6.43e6T^{2} \)
29 \( 1 + 4.76e3T + 2.05e7T^{2} \)
31 \( 1 - 6.72e3T + 2.86e7T^{2} \)
37 \( 1 - 3.09e3T + 6.93e7T^{2} \)
41 \( 1 - 1.66e4T + 1.15e8T^{2} \)
43 \( 1 - 1.66e4T + 1.47e8T^{2} \)
47 \( 1 + 5.19e3T + 2.29e8T^{2} \)
53 \( 1 + 4.99e3T + 4.18e8T^{2} \)
61 \( 1 + 4.12e4T + 8.44e8T^{2} \)
67 \( 1 - 4.13e4T + 1.35e9T^{2} \)
71 \( 1 + 3.79e3T + 1.80e9T^{2} \)
73 \( 1 + 1.55e4T + 2.07e9T^{2} \)
79 \( 1 - 6.47e3T + 3.07e9T^{2} \)
83 \( 1 + 2.03e4T + 3.93e9T^{2} \)
89 \( 1 - 3.55e4T + 5.58e9T^{2} \)
97 \( 1 + 1.84e4T + 8.58e9T^{2} \)
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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

−11.83256171692561757663039781064, −10.84470856492313200990950589655, −9.584901010732442112577685203886, −7.53530875317867733667913911507, −6.54534547003960793180106354875, −6.04049824818090472867320543782, −4.30191080897895582535347416008, −3.90668952206246203434348161009, −2.48553402020588528185654317513, 0, 2.48553402020588528185654317513, 3.90668952206246203434348161009, 4.30191080897895582535347416008, 6.04049824818090472867320543782, 6.54534547003960793180106354875, 7.53530875317867733667913911507, 9.584901010732442112577685203886, 10.84470856492313200990950589655, 11.83256171692561757663039781064

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