Properties

Label 2-2793-399.356-c0-0-0
Degree $2$
Conductor $2793$
Sign $0.671 - 0.740i$
Analytic cond. $1.39388$
Root an. cond. $1.18063$
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  + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.766 + 0.642i)4-s + (−0.939 − 0.342i)9-s + (0.5 + 0.866i)12-s + (0.326 + 1.85i)13-s + (0.173 − 0.984i)16-s + (−0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.766 + 1.32i)31-s + (0.939 − 0.342i)36-s + 1.96i·37-s + 1.87·39-s + (−0.266 − 0.223i)43-s + (−0.939 − 0.342i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8756466445\)
\(L(\frac12)\) \(\approx\) \(0.8756466445\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 + 0.984i)T \)
7 \( 1 \)
19 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.766 - 0.642i)T^{2} \)
5 \( 1 + (0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (0.766 + 0.642i)T^{2} \)
31 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.96iT - T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (0.766 + 0.642i)T^{2} \)
53 \( 1 + (0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.766 + 0.642i)T^{2} \)
61 \( 1 + (-0.826 - 0.984i)T + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.233 + 0.642i)T + (-0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.673 - 0.118i)T + (0.939 + 0.342i)T^{2} \)
79 \( 1 + (-1.26 - 0.223i)T + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.939 - 0.342i)T^{2} \)
97 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
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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

−8.789108078231855338346435219014, −8.395433708400424953023200059000, −7.72211384550965562831810656361, −6.67309757852154338332769288653, −6.42560153442304587196720201755, −5.13803011437355749258103386355, −4.28634591160026558106083749072, −3.48613729844348860055313343403, −2.42371076888873624515443201298, −1.35074426434797745995099839904, 0.59781308088336191949865761881, 2.36170627648345842047261750464, 3.44040312021017779341716224720, 4.14598495543983918908964801258, 5.09620598327130620208992632977, 5.54193307198223537649088243892, 6.30179447144839747937629523180, 7.66258762209651927777942920191, 8.295290488430041898843808910311, 9.034820468552344851215424127829

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