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Properties

Label 2-1407-1407.1319-c0-0-0
Degree $2$
Conductor $1407$
Sign $0.925 + 0.378i$
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  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯
L(s)  = 1  + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $0.925 + 0.378i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ 0.925 + 0.378i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.365655437\)
\(L(\frac12)\) \(\approx\) \(1.365655437\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.580 - 0.814i)T \)
7 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
good2 \( 1 + (-0.142 + 0.989i)T^{2} \)
5 \( 1 + (0.888 - 0.458i)T^{2} \)
11 \( 1 + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (-1.16 + 0.600i)T + (0.580 - 0.814i)T^{2} \)
17 \( 1 + (0.235 - 0.971i)T^{2} \)
19 \( 1 + (-0.143 - 0.124i)T + (0.142 + 0.989i)T^{2} \)
23 \( 1 + (0.654 + 0.755i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
37 \( 1 - 0.0951T + T^{2} \)
41 \( 1 + (-0.235 + 0.971i)T^{2} \)
43 \( 1 + (-1.07 + 0.153i)T + (0.959 - 0.281i)T^{2} \)
47 \( 1 + (-0.327 + 0.945i)T^{2} \)
53 \( 1 + (0.235 + 0.971i)T^{2} \)
59 \( 1 + (-0.995 + 0.0950i)T^{2} \)
61 \( 1 + (1.91 - 0.560i)T + (0.841 - 0.540i)T^{2} \)
71 \( 1 + (-0.723 + 0.690i)T^{2} \)
73 \( 1 + (1.34 - 1.40i)T + (-0.0475 - 0.998i)T^{2} \)
79 \( 1 + (-0.915 - 1.77i)T + (-0.580 + 0.814i)T^{2} \)
83 \( 1 + (0.0475 - 0.998i)T^{2} \)
89 \( 1 + (-0.327 - 0.945i)T^{2} \)
97 \( 1 + (-0.580 + 1.00i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.881015897810035803863165780698, −9.069101747719300265267699396625, −8.154262435546445845339174261133, −7.40663482207328874149709720429, −6.24031460469296276692130018724, −5.57078669819718937243750182116, −4.48278792555911361742751908304, −3.81845553794549652178629245485, −2.65940897839852724161540275496, −1.22831713221134968030026347555, 1.72166448917007778309369517735, 2.73564177481551775723597103750, 3.48452945053310921058703218116, 4.57877484337510394353819784972, 6.10405023411206937068844189145, 6.47394956410698699440949465613, 7.58762625485116832485378334498, 8.179104344986139503165865029420, 8.879569972089805198703996439288, 9.340092826008785885035860783214

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