↑

Properties

Label 2-3789-421.379-c0-0-0
Degree $2$
Conductor $3789$
Sign $0.684 + 0.729i$
Analytic cond. $1.89095$
Root an. cond. $1.37512$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.512 − 0.858i)4-s + (1.01 − 0.433i)7-s + (1.65 + 0.842i)13-s + (−0.473 − 0.880i)16-s + (0.132 − 0.647i)19-s + (0.473 − 0.880i)25-s + (0.147 − 1.09i)28-s + (−1.95 + 0.355i)31-s + (0.0862 + 1.27i)37-s + (−0.487 + 1.62i)43-s + (0.147 − 0.154i)49-s + (1.57 − 0.987i)52-s + (−0.128 − 0.0386i)61-s + (−0.998 − 0.0448i)64-s + (0.668 + 0.217i)67-s + ⋯
L(s)  = 1  + (0.512 − 0.858i)4-s + (1.01 − 0.433i)7-s + (1.65 + 0.842i)13-s + (−0.473 − 0.880i)16-s + (0.132 − 0.647i)19-s + (0.473 − 0.880i)25-s + (0.147 − 1.09i)28-s + (−1.95 + 0.355i)31-s + (0.0862 + 1.27i)37-s + (−0.487 + 1.62i)43-s + (0.147 − 0.154i)49-s + (1.57 − 0.987i)52-s + (−0.128 − 0.0386i)61-s + (−0.998 − 0.0448i)64-s + (0.668 + 0.217i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3789 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3789\)    =    \(3^{2} \cdot 421\)
Sign: $0.684 + 0.729i$
Analytic conductor: \(1.89095\)
Root analytic conductor: \(1.37512\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3789} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3789,\ (\ :0),\ 0.684 + 0.729i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.760790118\)
\(L(\frac12)\) \(\approx\) \(1.760790118\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
421 \( 1 + (-0.919 - 0.393i)T \)
good2 \( 1 + (-0.512 + 0.858i)T^{2} \)
5 \( 1 + (-0.473 + 0.880i)T^{2} \)
7 \( 1 + (-1.01 + 0.433i)T + (0.691 - 0.722i)T^{2} \)
11 \( 1 + (0.134 + 0.990i)T^{2} \)
13 \( 1 + (-1.65 - 0.842i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.995 + 0.0896i)T^{2} \)
19 \( 1 + (-0.132 + 0.647i)T + (-0.919 - 0.393i)T^{2} \)
23 \( 1 + (0.178 - 0.983i)T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 + (1.95 - 0.355i)T + (0.936 - 0.351i)T^{2} \)
37 \( 1 + (-0.0862 - 1.27i)T + (-0.990 + 0.134i)T^{2} \)
41 \( 1 + (0.919 + 0.393i)T^{2} \)
43 \( 1 + (0.487 - 1.62i)T + (-0.834 - 0.550i)T^{2} \)
47 \( 1 + (0.880 + 0.473i)T^{2} \)
53 \( 1 + (-0.722 - 0.691i)T^{2} \)
59 \( 1 + (-0.657 - 0.753i)T^{2} \)
61 \( 1 + (0.128 + 0.0386i)T + (0.834 + 0.550i)T^{2} \)
67 \( 1 + (-0.668 - 0.217i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-0.657 + 0.753i)T^{2} \)
73 \( 1 + (-0.309 - 0.0489i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (0.252 - 0.468i)T + (-0.550 - 0.834i)T^{2} \)
83 \( 1 + (0.657 + 0.753i)T^{2} \)
89 \( 1 + (-0.351 + 0.936i)T^{2} \)
97 \( 1 + (-1.09 + 1.65i)T + (-0.393 - 0.919i)T^{2} \)
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

−8.586764801443533081465716524185, −7.88044388203868420863992328697, −6.91641724729258171340275556510, −6.47232997357625818061613195300, −5.61941835632375168083664516827, −4.82090373441033559674038521547, −4.13535353940181195642691392695, −3.00711347599703476350714173744, −1.77932062202849464228089610576, −1.20313908223995428252130820698, 1.46812701111442731189217418730, 2.29659286245153001197778295793, 3.55490650472099966423895433593, 3.80655052971180060233387936487, 5.20767441484299494732124420633, 5.70645696035245342808635701467, 6.61823827076496111684080874189, 7.54902197420845627801329284417, 7.944404669198842538291567644759, 8.739444029532727949731865447490

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