Properties

Label 2-177-1.1-c7-0-4
Degree $2$
Conductor $177$
Sign $1$
Analytic cond. $55.2921$
Root an. cond. $7.43586$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3.93·2-s + 27·3-s − 112.·4-s − 305.·5-s + 106.·6-s − 1.40e3·7-s − 946.·8-s + 729·9-s − 1.20e3·10-s + 322.·11-s − 3.03e3·12-s − 6.38e3·13-s − 5.53e3·14-s − 8.25e3·15-s + 1.06e4·16-s − 1.48e4·17-s + 2.86e3·18-s − 1.27e4·19-s + 3.43e4·20-s − 3.79e4·21-s + 1.26e3·22-s − 1.73e4·23-s − 2.55e4·24-s + 1.52e4·25-s − 2.51e4·26-s + 1.96e4·27-s + 1.58e5·28-s + ⋯
L(s)  = 1  + 0.347·2-s + 0.577·3-s − 0.879·4-s − 1.09·5-s + 0.200·6-s − 1.55·7-s − 0.653·8-s + 0.333·9-s − 0.380·10-s + 0.0729·11-s − 0.507·12-s − 0.806·13-s − 0.538·14-s − 0.631·15-s + 0.652·16-s − 0.733·17-s + 0.115·18-s − 0.425·19-s + 0.961·20-s − 0.895·21-s + 0.0253·22-s − 0.296·23-s − 0.377·24-s + 0.195·25-s − 0.280·26-s + 0.192·27-s + 1.36·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $1$
Analytic conductor: \(55.2921\)
Root analytic conductor: \(7.43586\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6376650875\)
\(L(\frac12)\) \(\approx\) \(0.6376650875\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
59 \( 1 - 2.05e5T \)
good2 \( 1 - 3.93T + 128T^{2} \)
5 \( 1 + 305.T + 7.81e4T^{2} \)
7 \( 1 + 1.40e3T + 8.23e5T^{2} \)
11 \( 1 - 322.T + 1.94e7T^{2} \)
13 \( 1 + 6.38e3T + 6.27e7T^{2} \)
17 \( 1 + 1.48e4T + 4.10e8T^{2} \)
19 \( 1 + 1.27e4T + 8.93e8T^{2} \)
23 \( 1 + 1.73e4T + 3.40e9T^{2} \)
29 \( 1 + 6.86e4T + 1.72e10T^{2} \)
31 \( 1 - 1.53e5T + 2.75e10T^{2} \)
37 \( 1 - 1.98e5T + 9.49e10T^{2} \)
41 \( 1 - 4.19e5T + 1.94e11T^{2} \)
43 \( 1 - 1.04e5T + 2.71e11T^{2} \)
47 \( 1 + 7.38e5T + 5.06e11T^{2} \)
53 \( 1 - 2.48e5T + 1.17e12T^{2} \)
61 \( 1 - 9.15e5T + 3.14e12T^{2} \)
67 \( 1 + 8.28e5T + 6.06e12T^{2} \)
71 \( 1 + 3.02e5T + 9.09e12T^{2} \)
73 \( 1 + 3.36e6T + 1.10e13T^{2} \)
79 \( 1 + 6.08e5T + 1.92e13T^{2} \)
83 \( 1 + 3.11e6T + 2.71e13T^{2} \)
89 \( 1 + 8.70e6T + 4.42e13T^{2} \)
97 \( 1 + 1.80e6T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65930120914800275223122607731, −10.10300313788074030981995243751, −9.361427966614332037632600092469, −8.421096388297436496632737002672, −7.33201532981638774770527343924, −6.14236792989924801278744334186, −4.52441909139769438626611306879, −3.74747054919846346920659487175, −2.74429771129966258661547282486, −0.38451077005128205142020335772, 0.38451077005128205142020335772, 2.74429771129966258661547282486, 3.74747054919846346920659487175, 4.52441909139769438626611306879, 6.14236792989924801278744334186, 7.33201532981638774770527343924, 8.421096388297436496632737002672, 9.361427966614332037632600092469, 10.10300313788074030981995243751, 11.65930120914800275223122607731

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