↑

Properties

Label 2-1407-1407.1082-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.853 + 0.520i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
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  + (0.928 + 0.371i)3-s + (−0.888 − 0.458i)4-s + (−0.959 − 0.281i)7-s + (0.723 + 0.690i)9-s + (−0.654 − 0.755i)12-s + (0.627 − 1.81i)13-s + (0.580 + 0.814i)16-s + (1.34 − 1.28i)19-s + (−0.786 − 0.618i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (0.723 + 0.690i)28-s + (0.815 + 0.157i)31-s + (−0.327 − 0.945i)36-s + (−0.981 + 1.70i)37-s + ⋯
L(s)  = 1  + (0.928 + 0.371i)3-s + (−0.888 − 0.458i)4-s + (−0.959 − 0.281i)7-s + (0.723 + 0.690i)9-s + (−0.654 − 0.755i)12-s + (0.627 − 1.81i)13-s + (0.580 + 0.814i)16-s + (1.34 − 1.28i)19-s + (−0.786 − 0.618i)21-s + (0.981 − 0.189i)25-s + (0.415 + 0.909i)27-s + (0.723 + 0.690i)28-s + (0.815 + 0.157i)31-s + (−0.327 − 0.945i)36-s + (−0.981 + 1.70i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.853 + 0.520i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.853 + 0.520i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.163629034\)
\(L(\frac12)\) \(\approx\) \(1.163629034\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 - 0.371i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good2 \( 1 + (0.888 + 0.458i)T^{2} \)
5 \( 1 + (-0.981 + 0.189i)T^{2} \)
11 \( 1 + (-0.981 + 0.189i)T^{2} \)
13 \( 1 + (-0.627 + 1.81i)T + (-0.786 - 0.618i)T^{2} \)
17 \( 1 + (-0.415 - 0.909i)T^{2} \)
19 \( 1 + (-1.34 + 1.28i)T + (0.0475 - 0.998i)T^{2} \)
23 \( 1 + (-0.723 - 0.690i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.815 - 0.157i)T + (0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.981 - 1.70i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.580 + 0.814i)T^{2} \)
43 \( 1 + (1.10 + 0.708i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.995 - 0.0950i)T^{2} \)
59 \( 1 + (0.786 - 0.618i)T^{2} \)
61 \( 1 + (1.84 + 0.176i)T + (0.981 + 0.189i)T^{2} \)
71 \( 1 + (0.995 - 0.0950i)T^{2} \)
73 \( 1 + (-0.481 - 1.05i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.981 + 0.189i)T^{2} \)
89 \( 1 + (-0.723 + 0.690i)T^{2} \)
97 \( 1 + (-0.786 - 1.36i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.691572705464969766256542668096, −8.895162344668201745095811083946, −8.333775459610739160048861554084, −7.41381827653819450295508077362, −6.41108848118558730858583095346, −5.28568006078724410711279675922, −4.65780341014906032101540080814, −3.32396991597532803579940631019, −3.06475830124788027631720265435, −1.03422754971839484393639144782, 1.53762698851530422074182184814, 2.99613063082957119472777242131, 3.66626703284685967523122038422, 4.46727318736973282330911809253, 5.78186853194452554596997669804, 6.75625854235887799410748934938, 7.45385361985801022030540917961, 8.415666843045264635723887632821, 9.044399953516351832426517179894, 9.490918917448901542317092363451

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