Properties

Label 2-1407-1407.1343-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.374 - 0.927i$
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  + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)12-s + (1.25 − 1.45i)13-s + (0.415 + 0.909i)16-s + (0.557 + 1.89i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)25-s + (0.415 − 0.909i)27-s + (−0.142 − 0.989i)28-s + (0.544 + 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯
L(s)  = 1  + (−0.142 + 0.989i)3-s + (−0.841 − 0.540i)4-s + (0.654 + 0.755i)7-s + (−0.959 − 0.281i)9-s + (0.654 − 0.755i)12-s + (1.25 − 1.45i)13-s + (0.415 + 0.909i)16-s + (0.557 + 1.89i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)25-s + (0.415 − 0.909i)27-s + (−0.142 − 0.989i)28-s + (0.544 + 0.627i)31-s + (0.654 + 0.755i)36-s − 1.30·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.374 - 0.927i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9074109216\)
\(L(\frac12)\) \(\approx\) \(0.9074109216\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (-0.654 - 0.755i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
good2 \( 1 + (0.841 + 0.540i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
11 \( 1 + (-0.654 + 0.755i)T^{2} \)
13 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.557 - 1.89i)T + (-0.841 + 0.540i)T^{2} \)
23 \( 1 + (0.959 + 0.281i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.544 - 0.627i)T + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + 1.30T + T^{2} \)
41 \( 1 + (-0.415 + 0.909i)T^{2} \)
43 \( 1 + (-0.817 - 1.27i)T + (-0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.415 + 0.909i)T^{2} \)
59 \( 1 + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (-0.415 - 0.909i)T^{2} \)
73 \( 1 + (-1.65 - 0.755i)T + (0.654 + 0.755i)T^{2} \)
79 \( 1 + (1.14 + 0.989i)T + (0.142 + 0.989i)T^{2} \)
83 \( 1 + (-0.654 + 0.755i)T^{2} \)
89 \( 1 + (-0.959 + 0.281i)T^{2} \)
97 \( 1 - 0.284T + T^{2} \)
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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

−9.918947570342871182425181797139, −9.181504059789465519871426831739, −8.326294479133930101411586929821, −8.008400640261710763285599465999, −6.09961540625581400220054914566, −5.61284043889039460550507458339, −5.06086400267791338255890641877, −3.93073882355604430354982915011, −3.21982009106973289087698182176, −1.41930669145409160204404597874, 0.920323668437309628811690524111, 2.27039684019481787884667443255, 3.66686831873195779475968369947, 4.46924219630918436245466158375, 5.38298026253382592057943427027, 6.59560724593486866713159719639, 7.16449348662446643686371193302, 8.012896440954638443741848905292, 8.717218728507969329011085570798, 9.264334896863654628952781729949

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