Properties

Label 2-1407-1407.1271-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.260 + 0.965i$
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  + (0.723 − 0.690i)3-s + (0.580 − 0.814i)4-s + (0.841 − 0.540i)7-s + (0.0475 − 0.998i)9-s + (−0.142 − 0.989i)12-s + (−1.32 + 1.04i)13-s + (−0.327 − 0.945i)16-s + (0.0688 + 1.44i)19-s + (0.235 − 0.971i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.0475 − 0.998i)28-s + (−1.21 + 0.486i)31-s + (−0.786 − 0.618i)36-s + (−0.928 + 1.60i)37-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)3-s + (0.580 − 0.814i)4-s + (0.841 − 0.540i)7-s + (0.0475 − 0.998i)9-s + (−0.142 − 0.989i)12-s + (−1.32 + 1.04i)13-s + (−0.327 − 0.945i)16-s + (0.0688 + 1.44i)19-s + (0.235 − 0.971i)21-s + (0.928 + 0.371i)25-s + (−0.654 − 0.755i)27-s + (0.0475 − 0.998i)28-s + (−1.21 + 0.486i)31-s + (−0.786 − 0.618i)36-s + (−0.928 + 1.60i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.260 + 0.965i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.260 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.602737434\)
\(L(\frac12)\) \(\approx\) \(1.602737434\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.723 + 0.690i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
good2 \( 1 + (-0.580 + 0.814i)T^{2} \)
5 \( 1 + (-0.928 - 0.371i)T^{2} \)
11 \( 1 + (-0.928 - 0.371i)T^{2} \)
13 \( 1 + (1.32 - 1.04i)T + (0.235 - 0.971i)T^{2} \)
17 \( 1 + (0.654 + 0.755i)T^{2} \)
19 \( 1 + (-0.0688 - 1.44i)T + (-0.995 + 0.0950i)T^{2} \)
23 \( 1 + (-0.0475 + 0.998i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (1.21 - 0.486i)T + (0.723 - 0.690i)T^{2} \)
37 \( 1 + (0.928 - 1.60i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.327 - 0.945i)T^{2} \)
43 \( 1 + (0.118 - 0.258i)T + (-0.654 - 0.755i)T^{2} \)
47 \( 1 + (-0.841 - 0.540i)T^{2} \)
53 \( 1 + (-0.981 - 0.189i)T^{2} \)
59 \( 1 + (-0.235 - 0.971i)T^{2} \)
61 \( 1 + (-1.42 + 0.273i)T + (0.928 - 0.371i)T^{2} \)
71 \( 1 + (-0.981 - 0.189i)T^{2} \)
73 \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.264 + 1.83i)T + (-0.959 + 0.281i)T^{2} \)
83 \( 1 + (-0.928 - 0.371i)T^{2} \)
89 \( 1 + (-0.0475 - 0.998i)T^{2} \)
97 \( 1 + (0.235 + 0.408i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.641608585153903639608320793783, −8.736465614037908357495597130231, −7.82873753341963400956230512039, −7.11837346862947132691791629691, −6.62792026717703991879626228436, −5.44069915071064958697467493350, −4.57444909243038960959960517855, −3.32853980859802339555987946715, −2.05583911572033653276860938764, −1.44652488575096348734718058420, 2.24994484112791713038730641032, 2.71934854536296136627825984997, 3.82437883834550200292180062801, 4.86816361716712214835678509527, 5.49306636568383567415217093069, 7.08979048464832999857493964424, 7.51951153204532645304711722703, 8.426343022684034676252611809218, 8.915953645358095207101127598449, 9.838511727462246247589493675591

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