Properties

Label 2-177-1.1-c5-0-8
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  − 6.10·2-s + 9·3-s + 5.27·4-s − 41.8·5-s − 54.9·6-s + 208.·7-s + 163.·8-s + 81·9-s + 255.·10-s + 149.·11-s + 47.5·12-s − 952.·13-s − 1.27e3·14-s − 376.·15-s − 1.16e3·16-s + 1.07e3·17-s − 494.·18-s − 486.·19-s − 220.·20-s + 1.87e3·21-s − 910.·22-s + 2.19e3·23-s + 1.46e3·24-s − 1.37e3·25-s + 5.81e3·26-s + 729·27-s + 1.09e3·28-s + ⋯
L(s)  = 1  − 1.07·2-s + 0.577·3-s + 0.164·4-s − 0.747·5-s − 0.623·6-s + 1.60·7-s + 0.901·8-s + 0.333·9-s + 0.807·10-s + 0.371·11-s + 0.0952·12-s − 1.56·13-s − 1.73·14-s − 0.431·15-s − 1.13·16-s + 0.900·17-s − 0.359·18-s − 0.309·19-s − 0.123·20-s + 0.927·21-s − 0.401·22-s + 0.866·23-s + 0.520·24-s − 0.440·25-s + 1.68·26-s + 0.192·27-s + 0.265·28-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: \(0\)
Selberg data: \((2,\ 177,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.264824816\)
\(L(\frac12)\) \(\approx\) \(1.264824816\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.10T + 32T^{2} \)
5 \( 1 + 41.8T + 3.12e3T^{2} \)
7 \( 1 - 208.T + 1.68e4T^{2} \)
11 \( 1 - 149.T + 1.61e5T^{2} \)
13 \( 1 + 952.T + 3.71e5T^{2} \)
17 \( 1 - 1.07e3T + 1.41e6T^{2} \)
19 \( 1 + 486.T + 2.47e6T^{2} \)
23 \( 1 - 2.19e3T + 6.43e6T^{2} \)
29 \( 1 + 2.54e3T + 2.05e7T^{2} \)
31 \( 1 - 1.02e3T + 2.86e7T^{2} \)
37 \( 1 + 3.16e3T + 6.93e7T^{2} \)
41 \( 1 - 1.45e4T + 1.15e8T^{2} \)
43 \( 1 - 1.33e4T + 1.47e8T^{2} \)
47 \( 1 - 1.58e4T + 2.29e8T^{2} \)
53 \( 1 - 746.T + 4.18e8T^{2} \)
61 \( 1 - 3.16e4T + 8.44e8T^{2} \)
67 \( 1 + 3.69e4T + 1.35e9T^{2} \)
71 \( 1 - 6.70e4T + 1.80e9T^{2} \)
73 \( 1 - 1.54e4T + 2.07e9T^{2} \)
79 \( 1 - 4.57e4T + 3.07e9T^{2} \)
83 \( 1 - 1.64e4T + 3.93e9T^{2} \)
89 \( 1 + 2.84e4T + 5.58e9T^{2} \)
97 \( 1 - 7.99e4T + 8.58e9T^{2} \)
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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

−11.57614158117168556718453558247, −10.66815967795605534695140227173, −9.573055528351024616010617372114, −8.691286995673604855805974113495, −7.68057498949647701418881216670, −7.45704661438709648764973120736, −5.06993732018555113867176737875, −4.11075152170386331032581644027, −2.17107758571453173882624091204, −0.848080708115706036144041214948, 0.848080708115706036144041214948, 2.17107758571453173882624091204, 4.11075152170386331032581644027, 5.06993732018555113867176737875, 7.45704661438709648764973120736, 7.68057498949647701418881216670, 8.691286995673604855805974113495, 9.573055528351024616010617372114, 10.66815967795605534695140227173, 11.57614158117168556718453558247

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