Properties

Label 2-1407-1407.1370-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.756 + 0.654i$
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  + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s − 2·31-s + (0.499 − 0.866i)36-s − 37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s − 4-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)12-s + (−1 − 1.73i)13-s + 16-s − 1.73i·19-s + (0.499 − 0.866i)21-s + (−0.5 − 0.866i)25-s + 0.999·27-s + (−0.5 − 0.866i)28-s − 2·31-s + (0.499 − 0.866i)36-s − 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.756 + 0.654i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (1370, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.756 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4511849539\)
\(L(\frac12)\) \(\approx\) \(0.4511849539\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 - T + T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.278152642972630635155646574095, −8.568451644253397834571473565804, −7.85050085512153876118821882790, −7.16971415059782852804438322047, −5.90873029274508255560536639429, −5.25998666943246839174554869352, −4.76349462022880674944190266461, −3.15141896464246794970969996187, −2.10852158794518297775379770277, −0.40780221694120514307641377259, 1.66084184773786321973719852678, 3.75925045005516498802534279134, 4.00244691971236529086078066226, 5.00088982328977428210566181828, 5.59799972855046629081492137786, 6.84320675288176762772422012347, 7.67086241007659697492344939252, 8.691575439596810060323304319486, 9.425090439393583766033409931193, 9.946617523557472932548620705314

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