Properties

Label 2-1407-1407.1307-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.790 + 0.612i$
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

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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 + (1.37 + 0.0656i)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 + (1.37 + 0.0656i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.790 + 0.612i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.790 + 0.612i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8796805774\)
\(L(\frac12)\) \(\approx\) \(0.8796805774\)
\(L(1)\) 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 \( 1 + (-1.37 - 0.0656i)T + (0.995 + 0.0950i)T^{2} \)
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 \( 1 + (1.80 - 0.822i)T + (0.654 - 0.755i)T^{2} \)
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 \( 1 + (-1.42 - 1.23i)T + (0.142 + 0.989i)T^{2} \)
79 \( 1 + (-0.735 - 0.105i)T + (0.959 + 0.281i)T^{2} \)
83 \( 1 + (0.928 - 0.371i)T^{2} \)
89 \( 1 + (0.0475 - 0.998i)T^{2} \)
97 \( 1 + (0.235 - 0.408i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.595683170932136235986128372433, −8.837195744479720304758234880504, −8.156535440730043028096674961338, −7.08063223821630326034105889053, −6.28500128502988879974105150192, −5.45239402707700841591729028671, −4.97549809344763457768307853890, −3.82843346636712161233159466752, −2.01949919561065729356165337213, −1.19697645236610432906144764168, 1.11424581089159734210208830060, 3.35991363372218046766171502135, 3.68397166261606918073077082012, 5.10910305417689677366992879716, 5.15673627386463962323301416562, 6.60825461864936809949383791671, 7.47556548514151440929472346422, 8.380557713674138604211145217630, 8.910330556208401227395145421606, 9.964006968797019998264104050927

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