Properties

Label 2-177-1.1-c3-0-16
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  + 4.08·2-s + 3·3-s + 8.68·4-s − 7.45·5-s + 12.2·6-s + 34.0·7-s + 2.80·8-s + 9·9-s − 30.4·10-s + 30.1·11-s + 26.0·12-s − 5.30·13-s + 138.·14-s − 22.3·15-s − 58.0·16-s + 63.2·17-s + 36.7·18-s − 86.0·19-s − 64.7·20-s + 102.·21-s + 122.·22-s + 83.0·23-s + 8.40·24-s − 69.4·25-s − 21.6·26-s + 27·27-s + 295.·28-s + ⋯
L(s)  = 1  + 1.44·2-s + 0.577·3-s + 1.08·4-s − 0.666·5-s + 0.833·6-s + 1.83·7-s + 0.123·8-s + 0.333·9-s − 0.963·10-s + 0.825·11-s + 0.626·12-s − 0.113·13-s + 2.65·14-s − 0.384·15-s − 0.906·16-s + 0.902·17-s + 0.481·18-s − 1.03·19-s − 0.723·20-s + 1.06·21-s + 1.19·22-s + 0.752·23-s + 0.0714·24-s − 0.555·25-s − 0.163·26-s + 0.192·27-s + 1.99·28-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

\(L(2)\) \(\approx\) \(4.461356772\)
\(L(\frac12)\) \(\approx\) \(4.461356772\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.08T + 8T^{2} \)
5 \( 1 + 7.45T + 125T^{2} \)
7 \( 1 - 34.0T + 343T^{2} \)
11 \( 1 - 30.1T + 1.33e3T^{2} \)
13 \( 1 + 5.30T + 2.19e3T^{2} \)
17 \( 1 - 63.2T + 4.91e3T^{2} \)
19 \( 1 + 86.0T + 6.85e3T^{2} \)
23 \( 1 - 83.0T + 1.21e4T^{2} \)
29 \( 1 + 27.8T + 2.43e4T^{2} \)
31 \( 1 + 48.3T + 2.97e4T^{2} \)
37 \( 1 + 358.T + 5.06e4T^{2} \)
41 \( 1 + 139.T + 6.89e4T^{2} \)
43 \( 1 + 366.T + 7.95e4T^{2} \)
47 \( 1 + 130.T + 1.03e5T^{2} \)
53 \( 1 - 608.T + 1.48e5T^{2} \)
61 \( 1 + 361.T + 2.26e5T^{2} \)
67 \( 1 + 519.T + 3.00e5T^{2} \)
71 \( 1 + 733.T + 3.57e5T^{2} \)
73 \( 1 - 1.13e3T + 3.89e5T^{2} \)
79 \( 1 - 280.T + 4.93e5T^{2} \)
83 \( 1 - 239.T + 5.71e5T^{2} \)
89 \( 1 - 1.21e3T + 7.04e5T^{2} \)
97 \( 1 - 149.T + 9.12e5T^{2} \)
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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

−12.13572491008668159417956013683, −11.72401769192977958906747378300, −10.66433590378710891815346462752, −8.917495881411767829502829207921, −8.035164094152241681789540016139, −6.92190928331969405689673671612, −5.35860817900245858538110485146, −4.43475181838014569196791164359, −3.52697469299822199448925613870, −1.82446042836825657889543150674, 1.82446042836825657889543150674, 3.52697469299822199448925613870, 4.43475181838014569196791164359, 5.35860817900245858538110485146, 6.92190928331969405689673671612, 8.035164094152241681789540016139, 8.917495881411767829502829207921, 10.66433590378710891815346462752, 11.72401769192977958906747378300, 12.13572491008668159417956013683

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