↑

Properties

Label 2-5292-63.20-c1-0-21
Degree $2$
Conductor $5292$
Sign $0.516 + 0.856i$
Analytic cond. $42.2568$
Root an. cond. $6.50052$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.48 + 2.57i)5-s + (−4.09 − 2.36i)11-s + (−3.54 + 2.04i)13-s − 1.67·17-s − 4.91i·19-s + (4.25 − 2.45i)23-s + (−1.91 + 3.30i)25-s + (−0.238 − 0.137i)29-s + (−1.38 + 0.801i)31-s + 3.39·37-s + (3.55 + 6.15i)41-s + (5.22 − 9.05i)43-s + (−5.49 + 9.52i)47-s + 0.816i·53-s − 14.0i·55-s + ⋯
L(s)  = 1  + (0.664 + 1.15i)5-s + (−1.23 − 0.712i)11-s + (−0.981 + 0.566i)13-s − 0.405·17-s − 1.12i·19-s + (0.886 − 0.511i)23-s + (−0.382 + 0.661i)25-s + (−0.0442 − 0.0255i)29-s + (−0.249 + 0.143i)31-s + 0.557·37-s + (0.555 + 0.961i)41-s + (0.797 − 1.38i)43-s + (−0.802 + 1.38i)47-s + 0.112i·53-s − 1.89i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5292\)    =    \(2^{2} \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(42.2568\)
Root analytic conductor: \(6.50052\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5292} (4409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5292,\ (\ :1/2),\ 0.516 + 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.273554416\)
\(L(\frac12)\) \(\approx\) \(1.273554416\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.48 - 2.57i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.54 - 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + 4.91iT - 19T^{2} \)
23 \( 1 + (-4.25 + 2.45i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.238 + 0.137i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.38 - 0.801i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 + (-3.55 - 6.15i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.22 + 9.05i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.49 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.816iT - 53T^{2} \)
59 \( 1 + (-1.37 - 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.23 + 3.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.80 + 10.0i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 + (-6.15 + 10.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 6.99i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 + (-7.00 - 4.04i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.77516851343038991015337791283, −7.45592140811077595025713581275, −6.48439524854825305102312514891, −6.18078234507887195162158559548, −5.06266268684524356242273071015, −4.62055141874550973827803536001, −3.19932183451224277417408368627, −2.72278952721761402234924708029, −2.03701782414472576334406149176, −0.35731383778521728193838477511, 1.00372475667068164423013187044, 2.06498778163405170790357005304, 2.76300294160568464856110813915, 4.00109043022997323747643352072, 4.88673988132934318674761761302, 5.31555332776306901719831847016, 5.88107214618739699311336331051, 7.03160842511643418481709696814, 7.66167668606244593655759144969, 8.271131789444615534242073659719

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