↑

Properties

Label 2-177-59.58-c4-0-36
Degree $2$
Conductor $177$
Sign $-0.736 - 0.676i$
Analytic cond. $18.2964$
Root an. cond. $4.27743$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 4.85i·2-s − 5.19·3-s − 7.60·4-s − 12.5·5-s + 25.2i·6-s + 50.4·7-s − 40.7i·8-s + 27·9-s + 60.7i·10-s − 191. i·11-s + 39.5·12-s + 0.640i·13-s − 245. i·14-s + 64.9·15-s − 319.·16-s − 549.·17-s + ⋯
L(s)  = 1  − 1.21i·2-s − 0.577·3-s − 0.475·4-s − 0.500·5-s + 0.701i·6-s + 1.03·7-s − 0.637i·8-s + 0.333·9-s + 0.607i·10-s − 1.58i·11-s + 0.274·12-s + 0.00378i·13-s − 1.25i·14-s + 0.288·15-s − 1.24·16-s − 1.90·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $-0.736 - 0.676i$
Analytic conductor: \(18.2964\)
Root analytic conductor: \(4.27743\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :2),\ -0.736 - 0.676i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.8827584366\)
\(L(\frac12)\) \(\approx\) \(0.8827584366\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 5.19T \)
59 \( 1 + (2.56e3 + 2.35e3i)T \)
good2 \( 1 + 4.85iT - 16T^{2} \)
5 \( 1 + 12.5T + 625T^{2} \)
7 \( 1 - 50.4T + 2.40e3T^{2} \)
11 \( 1 + 191. iT - 1.46e4T^{2} \)
13 \( 1 - 0.640iT - 2.85e4T^{2} \)
17 \( 1 + 549.T + 8.35e4T^{2} \)
19 \( 1 - 524.T + 1.30e5T^{2} \)
23 \( 1 - 543. iT - 2.79e5T^{2} \)
29 \( 1 + 296.T + 7.07e5T^{2} \)
31 \( 1 + 1.07e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.16e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.12e3T + 2.82e6T^{2} \)
43 \( 1 + 1.52e3iT - 3.41e6T^{2} \)
47 \( 1 + 103. iT - 4.87e6T^{2} \)
53 \( 1 + 2.81e3T + 7.89e6T^{2} \)
61 \( 1 - 4.90e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.85e3iT - 2.01e7T^{2} \)
71 \( 1 - 5.82e3T + 2.54e7T^{2} \)
73 \( 1 + 6.75e3iT - 2.83e7T^{2} \)
79 \( 1 + 9.33e3T + 3.89e7T^{2} \)
83 \( 1 + 1.83e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.73e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.52e4iT - 8.85e7T^{2} \)
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

−11.42657774274364147221727730923, −10.96620474367873240343979069944, −9.666221475634662861681406974239, −8.489157963379956669022709543432, −7.27462194803255119237310434384, −5.84117925526934022373101696806, −4.47632696705323987792357381022, −3.30354995418928085646263146949, −1.68374632806477959159921473877, −0.34408059119288353359247913256, 1.96276178361754431393989884469, 4.49153071848316222834369384884, 5.09069105936178914753412651429, 6.55240309825810778810872814013, 7.32561858001620447958392100467, 8.151320607778477501484488863619, 9.384404728907593000729619105551, 10.86783180418615424347161559498, 11.57206485892255804475809115202, 12.54087975856227084353416752915

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