↑

Properties

Label 2-1424-356.183-c0-0-0
Degree $2$
Conductor $1424$
Sign $0.992 - 0.124i$
Analytic cond. $0.710668$
Root an. cond. $0.843011$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.304 + 0.474i)5-s + (0.281 − 0.959i)9-s + (0.574 + 0.767i)13-s + (0.627 − 0.544i)17-s + (0.283 + 0.620i)25-s + (1.83 − 0.398i)29-s + (−1.41 + 1.41i)37-s + (0.959 − 1.28i)41-s + (0.368 + 0.425i)45-s + (−0.909 + 0.415i)49-s + (1.29 + 0.186i)53-s + (−0.398 + 0.148i)61-s + (−0.538 + 0.0385i)65-s + (0.273 − 0.0801i)73-s + (−0.841 − 0.540i)81-s + ⋯
L(s)  = 1  + (−0.304 + 0.474i)5-s + (0.281 − 0.959i)9-s + (0.574 + 0.767i)13-s + (0.627 − 0.544i)17-s + (0.283 + 0.620i)25-s + (1.83 − 0.398i)29-s + (−1.41 + 1.41i)37-s + (0.959 − 1.28i)41-s + (0.368 + 0.425i)45-s + (−0.909 + 0.415i)49-s + (1.29 + 0.186i)53-s + (−0.398 + 0.148i)61-s + (−0.538 + 0.0385i)65-s + (0.273 − 0.0801i)73-s + (−0.841 − 0.540i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1424\)    =    \(2^{4} \cdot 89\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(0.710668\)
Root analytic conductor: \(0.843011\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1424} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1424,\ (\ :0),\ 0.992 - 0.124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.109085700\)
\(L(\frac12)\) \(\approx\) \(1.109085700\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 + (0.841 - 0.540i)T \)
good3 \( 1 + (-0.281 + 0.959i)T^{2} \)
5 \( 1 + (0.304 - 0.474i)T + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (0.909 - 0.415i)T^{2} \)
11 \( 1 + (-0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.574 - 0.767i)T + (-0.281 + 0.959i)T^{2} \)
17 \( 1 + (-0.627 + 0.544i)T + (0.142 - 0.989i)T^{2} \)
19 \( 1 + (-0.540 + 0.841i)T^{2} \)
23 \( 1 + (0.540 - 0.841i)T^{2} \)
29 \( 1 + (-1.83 + 0.398i)T + (0.909 - 0.415i)T^{2} \)
31 \( 1 + (0.540 + 0.841i)T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 + (-0.959 + 1.28i)T + (-0.281 - 0.959i)T^{2} \)
43 \( 1 + (-0.909 - 0.415i)T^{2} \)
47 \( 1 + (-0.959 + 0.281i)T^{2} \)
53 \( 1 + (-1.29 - 0.186i)T + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.281 - 0.959i)T^{2} \)
61 \( 1 + (0.398 - 0.148i)T + (0.755 - 0.654i)T^{2} \)
67 \( 1 + (0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.415 - 0.909i)T^{2} \)
73 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
79 \( 1 + (0.841 - 0.540i)T^{2} \)
83 \( 1 + (0.989 + 0.142i)T^{2} \)
97 \( 1 + (1.66 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
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

−9.749167156205042607447885274850, −8.968653970849591312574582136299, −8.219744521017655625909167358417, −7.09463471076677640643764825485, −6.69816850960380746116020946638, −5.72443884798063038877140853571, −4.58387912385787175925784208679, −3.67599151260641255117820138670, −2.84923202894841296994521971791, −1.26821178756102249703269645693, 1.23310033476632181331293654967, 2.62342257719319214647635010841, 3.76404051691202706251974806923, 4.73146462612577929723291386068, 5.46452175794068922153004859450, 6.43755670070663672985683005590, 7.46026499519195032209488610995, 8.242396666668869828646447181878, 8.637304327174503604530468264479, 9.864842722631609553095531069910

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