Properties

Label 2-1216-19.18-c2-0-69
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $33.1336$
Root an. cond. $5.75617$
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  − 2.82i·3-s + 5-s − 5·7-s + 0.999·9-s + 5·11-s − 16.9i·13-s − 2.82i·15-s − 25·17-s + 19·19-s + 14.1i·21-s + 10·23-s − 24·25-s − 28.2i·27-s + 42.4i·29-s − 42.4i·31-s + ⋯
L(s)  = 1  − 0.942i·3-s + 0.200·5-s − 0.714·7-s + 0.111·9-s + 0.454·11-s − 1.30i·13-s − 0.188i·15-s − 1.47·17-s + 19-s + 0.673i·21-s + 0.434·23-s − 0.959·25-s − 1.04i·27-s + 1.46i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-1$
Analytic conductor: \(33.1336\)
Root analytic conductor: \(5.75617\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1),\ -1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9977659696\)
\(L(\frac12)\) \(\approx\) \(0.9977659696\)
\(L(2)\) 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 \( 1 - 19T \)
good3 \( 1 + 2.82iT - 9T^{2} \)
5 \( 1 - T + 25T^{2} \)
7 \( 1 + 5T + 49T^{2} \)
11 \( 1 - 5T + 121T^{2} \)
13 \( 1 + 16.9iT - 169T^{2} \)
17 \( 1 + 25T + 289T^{2} \)
23 \( 1 - 10T + 529T^{2} \)
29 \( 1 - 42.4iT - 841T^{2} \)
31 \( 1 + 42.4iT - 961T^{2} \)
37 \( 1 + 25.4iT - 1.36e3T^{2} \)
41 \( 1 - 42.4iT - 1.68e3T^{2} \)
43 \( 1 - 5T + 1.84e3T^{2} \)
47 \( 1 + 5T + 2.20e3T^{2} \)
53 \( 1 + 25.4iT - 2.80e3T^{2} \)
59 \( 1 + 84.8iT - 3.48e3T^{2} \)
61 \( 1 + 95T + 3.72e3T^{2} \)
67 \( 1 + 110. iT - 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 25T + 5.32e3T^{2} \)
79 \( 1 - 42.4iT - 6.24e3T^{2} \)
83 \( 1 + 130T + 6.88e3T^{2} \)
89 \( 1 - 127. iT - 7.92e3T^{2} \)
97 \( 1 + 16.9iT - 9.40e3T^{2} \)
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

−9.248721698815674170275888841848, −8.126668284886386582934411439794, −7.44798880492121281146743714858, −6.63279488846686857183789810806, −6.02618774718221267691993011872, −4.94968162690077241625855272886, −3.69752102300707023834843720612, −2.67701922849005728523121880777, −1.51464852629502267264975763343, −0.28877735576627156860567067410, 1.61890187183794563666153029922, 2.95769784078299026426650093159, 4.07753423488268528162614588511, 4.54049235092238358804179232279, 5.74087437482306725756663167257, 6.66607873069851729454915424837, 7.27818463600010053571397630899, 8.714721162518727063888217908963, 9.271442523655579667178009052352, 9.808961051413855264507718860053

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