Properties

Label 2-177-1.1-c3-0-9
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $10.4433$
Root an. cond. $3.23161$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.26·2-s + 3·3-s + 10.1·4-s + 13.9·5-s − 12.7·6-s + 21.4·7-s − 9.31·8-s + 9·9-s − 59.6·10-s + 50.2·11-s + 30.5·12-s − 31.1·13-s − 91.5·14-s + 41.9·15-s − 41.7·16-s + 2.00·17-s − 38.3·18-s − 59.2·19-s + 142.·20-s + 64.4·21-s − 214.·22-s − 28.6·23-s − 27.9·24-s + 70.3·25-s + 133.·26-s + 27·27-s + 218.·28-s + ⋯
L(s)  = 1  − 1.50·2-s + 0.577·3-s + 1.27·4-s + 1.25·5-s − 0.870·6-s + 1.15·7-s − 0.411·8-s + 0.333·9-s − 1.88·10-s + 1.37·11-s + 0.735·12-s − 0.665·13-s − 1.74·14-s + 0.721·15-s − 0.652·16-s + 0.0286·17-s − 0.502·18-s − 0.715·19-s + 1.59·20-s + 0.669·21-s − 2.07·22-s − 0.259·23-s − 0.237·24-s + 0.562·25-s + 1.00·26-s + 0.192·27-s + 1.47·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(10.4433\)
Root analytic conductor: \(3.23161\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.451251488\)
\(L(\frac12)\) \(\approx\) \(1.451251488\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
59 \( 1 - 59T \)
good2 \( 1 + 4.26T + 8T^{2} \)
5 \( 1 - 13.9T + 125T^{2} \)
7 \( 1 - 21.4T + 343T^{2} \)
11 \( 1 - 50.2T + 1.33e3T^{2} \)
13 \( 1 + 31.1T + 2.19e3T^{2} \)
17 \( 1 - 2.00T + 4.91e3T^{2} \)
19 \( 1 + 59.2T + 6.85e3T^{2} \)
23 \( 1 + 28.6T + 1.21e4T^{2} \)
29 \( 1 - 92.9T + 2.43e4T^{2} \)
31 \( 1 + 88.6T + 2.97e4T^{2} \)
37 \( 1 - 231.T + 5.06e4T^{2} \)
41 \( 1 - 237.T + 6.89e4T^{2} \)
43 \( 1 - 29.7T + 7.95e4T^{2} \)
47 \( 1 + 433.T + 1.03e5T^{2} \)
53 \( 1 - 464.T + 1.48e5T^{2} \)
61 \( 1 + 328.T + 2.26e5T^{2} \)
67 \( 1 + 841.T + 3.00e5T^{2} \)
71 \( 1 - 46.3T + 3.57e5T^{2} \)
73 \( 1 + 738.T + 3.89e5T^{2} \)
79 \( 1 - 734.T + 4.93e5T^{2} \)
83 \( 1 - 353.T + 5.71e5T^{2} \)
89 \( 1 + 704.T + 7.04e5T^{2} \)
97 \( 1 - 1.47e3T + 9.12e5T^{2} \)
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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

−11.91593539623447509830196142464, −10.86320677247626768151343564535, −9.888343720279439886113863615028, −9.218684884810886153384767322619, −8.430459724710197471105149993981, −7.39342958644130048464243837513, −6.22430628089017199791472378192, −4.53098844311359287533723092040, −2.22294934577190646855413797929, −1.35041683845663168362340337737, 1.35041683845663168362340337737, 2.22294934577190646855413797929, 4.53098844311359287533723092040, 6.22430628089017199791472378192, 7.39342958644130048464243837513, 8.430459724710197471105149993981, 9.218684884810886153384767322619, 9.888343720279439886113863615028, 10.86320677247626768151343564535, 11.91593539623447509830196142464

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