Properties

Label 2-177-1.1-c3-0-6
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  − 0.0357·2-s + 3·3-s − 7.99·4-s − 7.80·5-s − 0.107·6-s + 4.90·7-s + 0.571·8-s + 9·9-s + 0.279·10-s + 37.9·11-s − 23.9·12-s + 27.2·13-s − 0.175·14-s − 23.4·15-s + 63.9·16-s + 63.3·17-s − 0.321·18-s + 97.9·19-s + 62.4·20-s + 14.7·21-s − 1.35·22-s + 9.58·23-s + 1.71·24-s − 64.0·25-s − 0.972·26-s + 27·27-s − 39.2·28-s + ⋯
L(s)  = 1  − 0.0126·2-s + 0.577·3-s − 0.999·4-s − 0.698·5-s − 0.00729·6-s + 0.264·7-s + 0.0252·8-s + 0.333·9-s + 0.00882·10-s + 1.04·11-s − 0.577·12-s + 0.580·13-s − 0.00334·14-s − 0.403·15-s + 0.999·16-s + 0.903·17-s − 0.00421·18-s + 1.18·19-s + 0.698·20-s + 0.152·21-s − 0.0131·22-s + 0.0868·23-s + 0.0145·24-s − 0.512·25-s − 0.00733·26-s + 0.192·27-s − 0.264·28-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

\(L(2)\) \(\approx\) \(1.649557639\)
\(L(\frac12)\) \(\approx\) \(1.649557639\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.0357T + 8T^{2} \)
5 \( 1 + 7.80T + 125T^{2} \)
7 \( 1 - 4.90T + 343T^{2} \)
11 \( 1 - 37.9T + 1.33e3T^{2} \)
13 \( 1 - 27.2T + 2.19e3T^{2} \)
17 \( 1 - 63.3T + 4.91e3T^{2} \)
19 \( 1 - 97.9T + 6.85e3T^{2} \)
23 \( 1 - 9.58T + 1.21e4T^{2} \)
29 \( 1 - 65.2T + 2.43e4T^{2} \)
31 \( 1 + 165.T + 2.97e4T^{2} \)
37 \( 1 - 215.T + 5.06e4T^{2} \)
41 \( 1 - 61.9T + 6.89e4T^{2} \)
43 \( 1 - 77.4T + 7.95e4T^{2} \)
47 \( 1 - 189.T + 1.03e5T^{2} \)
53 \( 1 - 558.T + 1.48e5T^{2} \)
61 \( 1 - 451.T + 2.26e5T^{2} \)
67 \( 1 + 730.T + 3.00e5T^{2} \)
71 \( 1 - 1.04e3T + 3.57e5T^{2} \)
73 \( 1 + 529.T + 3.89e5T^{2} \)
79 \( 1 + 1.00e3T + 4.93e5T^{2} \)
83 \( 1 + 393.T + 5.71e5T^{2} \)
89 \( 1 + 50.5T + 7.04e5T^{2} \)
97 \( 1 + 1.04e3T + 9.12e5T^{2} \)
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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

−12.25869342976133780538501383282, −11.39280411579479645202949987316, −9.960618572997652833976388168860, −9.130507397620017230457133962579, −8.228959727562557878744335020676, −7.34856797747743420430299454913, −5.65762029449133554802796713064, −4.24570465544780806227149436518, −3.42890203885916183776586359926, −1.08595591761035775567808517584, 1.08595591761035775567808517584, 3.42890203885916183776586359926, 4.24570465544780806227149436518, 5.65762029449133554802796713064, 7.34856797747743420430299454913, 8.228959727562557878744335020676, 9.130507397620017230457133962579, 9.960618572997652833976388168860, 11.39280411579479645202949987316, 12.25869342976133780538501383282

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