↑

Properties

Label 2-177-59.58-c10-0-46
Degree $2$
Conductor $177$
Sign $-0.696 - 0.717i$
Analytic cond. $112.458$
Root an. cond. $10.6046$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 54.8i·2-s − 140.·3-s − 1.98e3·4-s − 4.48e3·5-s + 7.69e3i·6-s − 2.73e4·7-s + 5.24e4i·8-s + 1.96e4·9-s + 2.45e5i·10-s − 2.14e4i·11-s + 2.77e5·12-s − 3.09e5i·13-s + 1.50e6i·14-s + 6.28e5·15-s + 8.46e5·16-s + 1.91e6·17-s + ⋯
L(s)  = 1  − 1.71i·2-s − 0.577·3-s − 1.93·4-s − 1.43·5-s + 0.988i·6-s − 1.62·7-s + 1.60i·8-s + 0.333·9-s + 2.45i·10-s − 0.133i·11-s + 1.11·12-s − 0.832i·13-s + 2.78i·14-s + 0.827·15-s + 0.807·16-s + 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(112.458\)
Root analytic conductor: \(10.6046\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :5),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.5741142719\)
\(L(\frac12)\) \(\approx\) \(0.5741142719\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 140.T \)
59 \( 1 + (-4.98e8 - 5.12e8i)T \)
good2 \( 1 + 54.8iT - 1.02e3T^{2} \)
5 \( 1 + 4.48e3T + 9.76e6T^{2} \)
7 \( 1 + 2.73e4T + 2.82e8T^{2} \)
11 \( 1 + 2.14e4iT - 2.59e10T^{2} \)
13 \( 1 + 3.09e5iT - 1.37e11T^{2} \)
17 \( 1 - 1.91e6T + 2.01e12T^{2} \)
19 \( 1 + 6.75e5T + 6.13e12T^{2} \)
23 \( 1 + 4.00e6iT - 4.14e13T^{2} \)
29 \( 1 - 1.18e7T + 4.20e14T^{2} \)
31 \( 1 + 1.59e7iT - 8.19e14T^{2} \)
37 \( 1 + 7.63e7iT - 4.80e15T^{2} \)
41 \( 1 - 4.81e7T + 1.34e16T^{2} \)
43 \( 1 - 1.47e8iT - 2.16e16T^{2} \)
47 \( 1 + 8.31e7iT - 5.25e16T^{2} \)
53 \( 1 - 5.12e8T + 1.74e17T^{2} \)
61 \( 1 - 2.96e8iT - 7.13e17T^{2} \)
67 \( 1 - 5.57e8iT - 1.82e18T^{2} \)
71 \( 1 + 6.33e8T + 3.25e18T^{2} \)
73 \( 1 + 6.09e8iT - 4.29e18T^{2} \)
79 \( 1 + 1.56e9T + 9.46e18T^{2} \)
83 \( 1 + 1.79e9iT - 1.55e19T^{2} \)
89 \( 1 - 9.80e9iT - 3.11e19T^{2} \)
97 \( 1 + 7.87e9iT - 7.37e19T^{2} \)
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

−10.38185172720290475145179247803, −9.729755941482671814226819552209, −8.472279482913222775966330327815, −7.22285059617528767668900977557, −5.77834853152998650409663570514, −4.27804309400788622288208317491, −3.50267136524318077137719150514, −2.74998802064407223673833549070, −0.814261250409951898108829194614, −0.32996104969835984210106787013, 0.63841796344587186878722264411, 3.39138498011857179017439720343, 4.28804097988121038564554475144, 5.49957793593531988871859956549, 6.56448151506260528853372556413, 7.13339003775331424110654280351, 8.068744910191859103150986930369, 9.176618468824249509232533689862, 10.17195641013243306918749161770, 11.71854564068772717795331738756

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