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Properties

Label 2-2793-399.314-c0-0-1
Degree $2$
Conductor $2793$
Sign $0.349 + 0.937i$
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  + (0.766 − 0.642i)3-s + (0.939 − 0.342i)4-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)12-s + (−0.266 − 0.223i)13-s + (0.766 − 0.642i)16-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (−0.500 − 0.866i)27-s + (−0.939 + 1.62i)31-s + (−0.173 − 0.984i)36-s + 1.28i·37-s − 0.347·39-s + (1.43 + 0.524i)43-s + (0.173 − 0.984i)48-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.939 − 0.342i)4-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)12-s + (−0.266 − 0.223i)13-s + (0.766 − 0.642i)16-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (−0.500 − 0.866i)27-s + (−0.939 + 1.62i)31-s + (−0.173 − 0.984i)36-s + 1.28i·37-s − 0.347·39-s + (1.43 + 0.524i)43-s + (0.173 − 0.984i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2793\)    =    \(3 \cdot 7^{2} \cdot 19\)
Sign: $0.349 + 0.937i$
Analytic conductor: \(1.39388\)
Root analytic conductor: \(1.18063\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2793} (1910, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2793,\ (\ :0),\ 0.349 + 0.937i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.937142352\)
\(L(\frac12)\) \(\approx\) \(1.937142352\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 \)
19 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.939 + 0.342i)T^{2} \)
5 \( 1 + (0.766 + 0.642i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - 1.28iT - T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
47 \( 1 + (-0.939 - 0.342i)T^{2} \)
53 \( 1 + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.233 - 0.642i)T + (-0.766 + 0.642i)T^{2} \)
67 \( 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
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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

−8.699562342697277958169620877922, −8.039734006817836410426993303086, −7.23084243871308853287342651772, −6.74142786026584923520610596649, −6.00917347821099060174841011530, −5.04716184733886218186966558161, −3.85947627675494708454267819506, −2.86113413774530266778410411599, −2.23856253907122666516986393406, −1.16762126995952942506135026883, 1.91362060488317683748138259465, 2.44926523662053302857148600610, 3.68569873832408396817612834463, 4.00147513394692330703234632209, 5.32942311006452604177710729938, 6.03451959409928460416989555727, 7.07933659208487522002044404643, 7.74545341111830075819974387600, 8.204290215789956263068567153476, 9.284587045322667655575964261810

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