Properties

Label 2-177-1.1-c5-0-44
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  + 7.66·2-s + 9·3-s + 26.8·4-s − 109.·5-s + 69.0·6-s + 156.·7-s − 39.7·8-s + 81·9-s − 842.·10-s − 426.·11-s + 241.·12-s − 123.·13-s + 1.19e3·14-s − 988.·15-s − 1.16e3·16-s − 852.·17-s + 621.·18-s + 232.·19-s − 2.94e3·20-s + 1.40e3·21-s − 3.27e3·22-s − 3.64e3·23-s − 358.·24-s + 8.93e3·25-s − 950.·26-s + 729·27-s + 4.19e3·28-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.577·3-s + 0.837·4-s − 1.96·5-s + 0.782·6-s + 1.20·7-s − 0.219·8-s + 0.333·9-s − 2.66·10-s − 1.06·11-s + 0.483·12-s − 0.203·13-s + 1.63·14-s − 1.13·15-s − 1.13·16-s − 0.715·17-s + 0.451·18-s + 0.147·19-s − 1.64·20-s + 0.696·21-s − 1.44·22-s − 1.43·23-s − 0.126·24-s + 2.85·25-s − 0.275·26-s + 0.192·27-s + 1.01·28-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 \( 1 - 7.66T + 32T^{2} \)
5 \( 1 + 109.T + 3.12e3T^{2} \)
7 \( 1 - 156.T + 1.68e4T^{2} \)
11 \( 1 + 426.T + 1.61e5T^{2} \)
13 \( 1 + 123.T + 3.71e5T^{2} \)
17 \( 1 + 852.T + 1.41e6T^{2} \)
19 \( 1 - 232.T + 2.47e6T^{2} \)
23 \( 1 + 3.64e3T + 6.43e6T^{2} \)
29 \( 1 + 8.17e3T + 2.05e7T^{2} \)
31 \( 1 + 9.35e3T + 2.86e7T^{2} \)
37 \( 1 - 6.31e3T + 6.93e7T^{2} \)
41 \( 1 - 1.30e4T + 1.15e8T^{2} \)
43 \( 1 - 2.33e4T + 1.47e8T^{2} \)
47 \( 1 + 8.86e3T + 2.29e8T^{2} \)
53 \( 1 - 2.30e4T + 4.18e8T^{2} \)
61 \( 1 - 1.36e4T + 8.44e8T^{2} \)
67 \( 1 + 1.42e3T + 1.35e9T^{2} \)
71 \( 1 - 3.45e4T + 1.80e9T^{2} \)
73 \( 1 + 5.01e4T + 2.07e9T^{2} \)
79 \( 1 + 6.97e4T + 3.07e9T^{2} \)
83 \( 1 - 4.98e4T + 3.93e9T^{2} \)
89 \( 1 + 5.07e4T + 5.58e9T^{2} \)
97 \( 1 - 2.19e4T + 8.58e9T^{2} \)
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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

−11.44608214516480729728436362633, −10.95844993477036685560307926854, −8.960384777370689026828636767186, −7.86424574590455072330140316334, −7.39068443132225472405953006552, −5.46992118620934781966119167162, −4.36042903086877353026528197734, −3.81018393958993599091467508011, −2.40083319562865224653682549025, 0, 2.40083319562865224653682549025, 3.81018393958993599091467508011, 4.36042903086877353026528197734, 5.46992118620934781966119167162, 7.39068443132225472405953006552, 7.86424574590455072330140316334, 8.960384777370689026828636767186, 10.95844993477036685560307926854, 11.44608214516480729728436362633

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