Properties

Label 2-177-3.2-c2-0-25
Degree $2$
Conductor $177$
Sign $0.722 - 0.691i$
Analytic cond. $4.82290$
Root an. cond. $2.19611$
Motivic weight $2$
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  + 1.50i·2-s + (2.07 + 2.16i)3-s + 1.73·4-s − 9.35i·5-s + (−3.26 + 3.12i)6-s + 7.09·7-s + 8.63i·8-s + (−0.390 + 8.99i)9-s + 14.0·10-s − 19.6i·11-s + (3.60 + 3.76i)12-s + 5.28·13-s + 10.6i·14-s + (20.2 − 19.4i)15-s − 6.03·16-s + 18.6i·17-s + ⋯
L(s)  = 1  + 0.752i·2-s + (0.691 + 0.722i)3-s + 0.434·4-s − 1.87i·5-s + (−0.543 + 0.520i)6-s + 1.01·7-s + 1.07i·8-s + (−0.0434 + 0.999i)9-s + 1.40·10-s − 1.78i·11-s + (0.300 + 0.313i)12-s + 0.406·13-s + 0.761i·14-s + (1.35 − 1.29i)15-s − 0.377·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(177\)    =    \(3 \cdot 59\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(4.82290\)
Root analytic conductor: \(2.19611\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{177} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 177,\ (\ :1),\ 0.722 - 0.691i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.09086 + 0.839585i\)
\(L(\frac12)\) \(\approx\) \(2.09086 + 0.839585i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.07 - 2.16i)T \)
59 \( 1 - 7.68iT \)
good2 \( 1 - 1.50iT - 4T^{2} \)
5 \( 1 + 9.35iT - 25T^{2} \)
7 \( 1 - 7.09T + 49T^{2} \)
11 \( 1 + 19.6iT - 121T^{2} \)
13 \( 1 - 5.28T + 169T^{2} \)
17 \( 1 - 18.6iT - 289T^{2} \)
19 \( 1 + 28.5T + 361T^{2} \)
23 \( 1 - 22.2iT - 529T^{2} \)
29 \( 1 - 17.6iT - 841T^{2} \)
31 \( 1 + 8.65T + 961T^{2} \)
37 \( 1 + 2.64T + 1.36e3T^{2} \)
41 \( 1 + 43.6iT - 1.68e3T^{2} \)
43 \( 1 - 14.7T + 1.84e3T^{2} \)
47 \( 1 - 43.3iT - 2.20e3T^{2} \)
53 \( 1 + 30.3iT - 2.80e3T^{2} \)
61 \( 1 - 67.1T + 3.72e3T^{2} \)
67 \( 1 - 33.3T + 4.48e3T^{2} \)
71 \( 1 + 61.7iT - 5.04e3T^{2} \)
73 \( 1 - 41.7T + 5.32e3T^{2} \)
79 \( 1 + 16.0T + 6.24e3T^{2} \)
83 \( 1 - 50.6iT - 6.88e3T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 - 58.4T + 9.40e3T^{2} \)
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

−12.79172153776746473864278487396, −11.40969851593449288520391836804, −10.71853784987376623353873541828, −8.956147894258289668192314049909, −8.436632608661283027985241587547, −7.977007626584883084426923859117, −5.90184495928397564042022693101, −5.12981307717738310303143190986, −3.89270773566592383821701842571, −1.71552269975251219092506566645, 2.01202716917768920185986374555, 2.60711218810247408308484730819, 4.09467560010694256891805266694, 6.54016933776657837448371825008, 7.05096051311576193890387754760, 7.974252326273349359826594072108, 9.656223321877346403585225609409, 10.52914057504200907077595588397, 11.36665191028513018614141417149, 12.14636297932864970120950488507

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