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Properties

Label 2-1407-1407.209-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.982 - 0.184i$
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  + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)4-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)12-s + (−0.118 − 0.258i)13-s + (0.841 + 0.540i)16-s + (1.49 − 0.215i)19-s + (0.959 − 0.281i)21-s + (0.415 + 0.909i)25-s + (0.841 − 0.540i)27-s + (−0.654 + 0.755i)28-s + (−0.698 + 1.53i)31-s + (−0.415 + 0.909i)36-s + 0.830·37-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)3-s + (0.959 + 0.281i)4-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (−0.415 − 0.909i)12-s + (−0.118 − 0.258i)13-s + (0.841 + 0.540i)16-s + (1.49 − 0.215i)19-s + (0.959 − 0.281i)21-s + (0.415 + 0.909i)25-s + (0.841 − 0.540i)27-s + (−0.654 + 0.755i)28-s + (−0.698 + 1.53i)31-s + (−0.415 + 0.909i)36-s + 0.830·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.982 - 0.184i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.982 - 0.184i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.070738808\)
\(L(\frac12)\) \(\approx\) \(1.070738808\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
good2 \( 1 + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (0.841 - 0.540i)T^{2} \)
19 \( 1 + (-1.49 + 0.215i)T + (0.959 - 0.281i)T^{2} \)
23 \( 1 + (0.142 - 0.989i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.698 - 1.53i)T + (-0.654 - 0.755i)T^{2} \)
37 \( 1 - 0.830T + T^{2} \)
41 \( 1 + (-0.841 + 0.540i)T^{2} \)
43 \( 1 + (0.512 + 1.74i)T + (-0.841 + 0.540i)T^{2} \)
47 \( 1 + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (0.841 + 0.540i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-1.10 + 0.708i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.584 - 0.909i)T + (-0.415 + 0.909i)T^{2} \)
79 \( 1 + (1.65 - 0.755i)T + (0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (-0.142 - 0.989i)T^{2} \)
97 \( 1 - 1.30T + T^{2} \)
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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

−9.871616166090528633376339936037, −8.859708398498477769564834660339, −7.977665718891066119408230047997, −7.13124984941074124404081513146, −6.69460848823332059221087108784, −5.60395142622272806544543704795, −5.24186821319888631428196176769, −3.41151415044487337585598276249, −2.58849950298346789797228197834, −1.47456704261035196302043683270, 1.08179550838694812936168018891, 2.77105033035002488881082334720, 3.73929145990956878398268862306, 4.68013996525160038674554342157, 5.71774271597366108765387168687, 6.36545682489697003631876101397, 7.16958798540012604122757404967, 7.904335288148313448718594045504, 9.319402249613171682977404185515, 9.912446548517031667480082908782

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