Properties

Label 2-177-1.1-c5-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.69·2-s − 9·3-s − 24.7·4-s − 80.3·5-s − 24.2·6-s − 162.·7-s − 153.·8-s + 81·9-s − 216.·10-s − 426.·11-s + 222.·12-s − 106.·13-s − 437.·14-s + 723.·15-s + 377.·16-s − 864.·17-s + 218.·18-s − 676.·19-s + 1.98e3·20-s + 1.45e3·21-s − 1.15e3·22-s + 783.·23-s + 1.37e3·24-s + 3.33e3·25-s − 286.·26-s − 729·27-s + 4.00e3·28-s + ⋯
L(s)  = 1  + 0.477·2-s − 0.577·3-s − 0.772·4-s − 1.43·5-s − 0.275·6-s − 1.25·7-s − 0.845·8-s + 0.333·9-s − 0.685·10-s − 1.06·11-s + 0.445·12-s − 0.174·13-s − 0.596·14-s + 0.829·15-s + 0.368·16-s − 0.725·17-s + 0.159·18-s − 0.429·19-s + 1.10·20-s + 0.721·21-s − 0.507·22-s + 0.308·23-s + 0.488·24-s + 1.06·25-s − 0.0832·26-s − 0.192·27-s + 0.965·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.07901174122\)
\(L(\frac12)\) \(\approx\) \(0.07901174122\)
\(L(\frac{7}{2})\) 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 \( 1 - 2.69T + 32T^{2} \)
5 \( 1 + 80.3T + 3.12e3T^{2} \)
7 \( 1 + 162.T + 1.68e4T^{2} \)
11 \( 1 + 426.T + 1.61e5T^{2} \)
13 \( 1 + 106.T + 3.71e5T^{2} \)
17 \( 1 + 864.T + 1.41e6T^{2} \)
19 \( 1 + 676.T + 2.47e6T^{2} \)
23 \( 1 - 783.T + 6.43e6T^{2} \)
29 \( 1 + 2.85e3T + 2.05e7T^{2} \)
31 \( 1 + 1.90e3T + 2.86e7T^{2} \)
37 \( 1 - 568.T + 6.93e7T^{2} \)
41 \( 1 + 5.53e3T + 1.15e8T^{2} \)
43 \( 1 + 6.90e3T + 1.47e8T^{2} \)
47 \( 1 + 2.90e3T + 2.29e8T^{2} \)
53 \( 1 + 2.92e4T + 4.18e8T^{2} \)
61 \( 1 - 1.52e4T + 8.44e8T^{2} \)
67 \( 1 + 3.75e4T + 1.35e9T^{2} \)
71 \( 1 - 3.24e4T + 1.80e9T^{2} \)
73 \( 1 + 6.01e4T + 2.07e9T^{2} \)
79 \( 1 + 5.42e4T + 3.07e9T^{2} \)
83 \( 1 - 5.92e4T + 3.93e9T^{2} \)
89 \( 1 + 5.81e4T + 5.58e9T^{2} \)
97 \( 1 - 1.74e5T + 8.58e9T^{2} \)
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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

−12.00217969574257985846035930941, −10.96281745651804983192577633747, −9.866729976217223704794199970350, −8.725238160242335239415238746130, −7.59932051067398613805544350884, −6.42262259457811725117705574659, −5.13717959438700368534521021678, −4.11106603543875836557620626645, −3.10935645830598113267951476901, −0.16061467988902900320233417912, 0.16061467988902900320233417912, 3.10935645830598113267951476901, 4.11106603543875836557620626645, 5.13717959438700368534521021678, 6.42262259457811725117705574659, 7.59932051067398613805544350884, 8.725238160242335239415238746130, 9.866729976217223704794199970350, 10.96281745651804983192577633747, 12.00217969574257985846035930941

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