↑

Properties

Label 2-608-19.18-c2-0-1
Degree $2$
Conductor $608$
Sign $0.719 - 0.694i$
Analytic cond. $16.5668$
Root an. cond. $4.07023$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.94i·3-s − 7.51·5-s − 11.4·7-s − 6.58·9-s − 14.4·11-s − 23.9i·13-s + 29.6i·15-s + 26.8·17-s + (13.2 + 13.6i)19-s + 45.1i·21-s + 0.456·23-s + 31.4·25-s − 9.52i·27-s + 7.85i·29-s + 16.0i·31-s + ⋯
L(s)  = 1  − 1.31i·3-s − 1.50·5-s − 1.63·7-s − 0.731·9-s − 1.31·11-s − 1.84i·13-s + 1.97i·15-s + 1.57·17-s + (0.694 + 0.719i)19-s + 2.14i·21-s + 0.0198·23-s + 1.25·25-s − 0.352i·27-s + 0.270i·29-s + 0.519i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.719 - 0.694i$
Analytic conductor: \(16.5668\)
Root analytic conductor: \(4.07023\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1),\ 0.719 - 0.694i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2157249326\)
\(L(\frac12)\) \(\approx\) \(0.2157249326\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (-13.2 - 13.6i)T \)
good3 \( 1 + 3.94iT - 9T^{2} \)
5 \( 1 + 7.51T + 25T^{2} \)
7 \( 1 + 11.4T + 49T^{2} \)
11 \( 1 + 14.4T + 121T^{2} \)
13 \( 1 + 23.9iT - 169T^{2} \)
17 \( 1 - 26.8T + 289T^{2} \)
23 \( 1 - 0.456T + 529T^{2} \)
29 \( 1 - 7.85iT - 841T^{2} \)
31 \( 1 - 16.0iT - 961T^{2} \)
37 \( 1 - 12.5iT - 1.36e3T^{2} \)
41 \( 1 - 48.5iT - 1.68e3T^{2} \)
43 \( 1 + 30.5T + 1.84e3T^{2} \)
47 \( 1 + 44.3T + 2.20e3T^{2} \)
53 \( 1 + 48.7iT - 2.80e3T^{2} \)
59 \( 1 - 43.7iT - 3.48e3T^{2} \)
61 \( 1 - 3.64T + 3.72e3T^{2} \)
67 \( 1 + 75.4iT - 4.48e3T^{2} \)
71 \( 1 - 19.4iT - 5.04e3T^{2} \)
73 \( 1 + 83.0T + 5.32e3T^{2} \)
79 \( 1 + 73.9iT - 6.24e3T^{2} \)
83 \( 1 - 4.27T + 6.88e3T^{2} \)
89 \( 1 - 105. iT - 7.92e3T^{2} \)
97 \( 1 + 177. iT - 9.40e3T^{2} \)
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

−10.41971114371143218386116242680, −9.921058521793265999255533823053, −8.233273332017095237030922860262, −7.82630820617883831602277327831, −7.26645506133303001179617817694, −6.16814114521832532171417713278, −5.23469769545757249121185784701, −3.30778759047001706073470165536, −3.08977200972810937919996657429, −0.858650816866717096316850293837, 0.11288936666889093029937205909, 2.98853461336825110978250799668, 3.68244821701086017835560278715, 4.47406200806519431505521209768, 5.53773410257378591088338117921, 6.90266779840636898063625421784, 7.63394686271430890924040872785, 8.816375431687296458956957757443, 9.628562441135841070373998340273, 10.15488980712967097531683444069

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