Properties

Label 2-177-1.1-c3-0-22
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  + 4.06·2-s + 3·3-s + 8.49·4-s + 16.2·5-s + 12.1·6-s + 6.77·7-s + 2.00·8-s + 9·9-s + 66.0·10-s − 16.5·11-s + 25.4·12-s − 82.3·13-s + 27.4·14-s + 48.8·15-s − 59.7·16-s + 6.48·17-s + 36.5·18-s + 151.·19-s + 138.·20-s + 20.3·21-s − 67.3·22-s − 154.·23-s + 6.02·24-s + 139.·25-s − 334.·26-s + 27·27-s + 57.5·28-s + ⋯
L(s)  = 1  + 1.43·2-s + 0.577·3-s + 1.06·4-s + 1.45·5-s + 0.829·6-s + 0.365·7-s + 0.0888·8-s + 0.333·9-s + 2.08·10-s − 0.454·11-s + 0.613·12-s − 1.75·13-s + 0.524·14-s + 0.840·15-s − 0.934·16-s + 0.0924·17-s + 0.478·18-s + 1.82·19-s + 1.54·20-s + 0.211·21-s − 0.652·22-s − 1.39·23-s + 0.0512·24-s + 1.11·25-s − 2.52·26-s + 0.192·27-s + 0.388·28-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

\(L(2)\) \(\approx\) \(4.915283867\)
\(L(\frac12)\) \(\approx\) \(4.915283867\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.06T + 8T^{2} \)
5 \( 1 - 16.2T + 125T^{2} \)
7 \( 1 - 6.77T + 343T^{2} \)
11 \( 1 + 16.5T + 1.33e3T^{2} \)
13 \( 1 + 82.3T + 2.19e3T^{2} \)
17 \( 1 - 6.48T + 4.91e3T^{2} \)
19 \( 1 - 151.T + 6.85e3T^{2} \)
23 \( 1 + 154.T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 + 79.4T + 2.97e4T^{2} \)
37 \( 1 - 414.T + 5.06e4T^{2} \)
41 \( 1 + 521.T + 6.89e4T^{2} \)
43 \( 1 + 265.T + 7.95e4T^{2} \)
47 \( 1 - 31.6T + 1.03e5T^{2} \)
53 \( 1 + 256.T + 1.48e5T^{2} \)
61 \( 1 + 68.7T + 2.26e5T^{2} \)
67 \( 1 - 198.T + 3.00e5T^{2} \)
71 \( 1 - 725.T + 3.57e5T^{2} \)
73 \( 1 - 571.T + 3.89e5T^{2} \)
79 \( 1 + 304.T + 4.93e5T^{2} \)
83 \( 1 - 731.T + 5.71e5T^{2} \)
89 \( 1 + 694.T + 7.04e5T^{2} \)
97 \( 1 + 597.T + 9.12e5T^{2} \)
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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

−12.51310727582915812542654095443, −11.69239756715870085483456773716, −10.01273482477152537854470711858, −9.567770104936104827031500839129, −7.937323650520057998937883583120, −6.65399104232796588515325085605, −5.44117438654571863888058235555, −4.78175984910134312931435393127, −3.05264810605709453813499907702, −2.07608455963124982518416107762, 2.07608455963124982518416107762, 3.05264810605709453813499907702, 4.78175984910134312931435393127, 5.44117438654571863888058235555, 6.65399104232796588515325085605, 7.937323650520057998937883583120, 9.567770104936104827031500839129, 10.01273482477152537854470711858, 11.69239756715870085483456773716, 12.51310727582915812542654095443

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