↑

Properties

Label 2-21e2-7.4-c3-0-8
Degree $2$
Conductor $441$
Sign $0.991 - 0.126i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 2.29i)2-s + (0.5 − 0.866i)4-s − 23.8·8-s + (−13.2 + 22.9i)11-s + (27.5 + 47.6i)16-s + 70·22-s + (108. + 187. i)23-s + (62.5 − 108. i)25-s − 264.·29-s + (−22.4 + 38.9i)32-s + (225 + 389. i)37-s + 180·43-s + (13.2 + 22.9i)44-s + (287 − 497. i)46-s − 330.·50-s + ⋯
L(s)  = 1  + (−0.467 − 0.810i)2-s + (0.0625 − 0.108i)4-s − 1.05·8-s + (−0.362 + 0.628i)11-s + (0.429 + 0.744i)16-s + 0.678·22-s + (0.983 + 1.70i)23-s + (0.5 − 0.866i)25-s − 1.69·29-s + (−0.124 + 0.215i)32-s + (0.999 + 1.73i)37-s + 0.638·43-s + (0.0453 + 0.0785i)44-s + (0.919 − 1.59i)46-s − 0.935·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.991 - 0.126i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.072670903\)
\(L(\frac12)\) \(\approx\) \(1.072670903\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (1.32 + 2.29i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (13.2 - 22.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 2.19e3T^{2} \)
17 \( 1 + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-108. - 187. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 264.T + 2.43e4T^{2} \)
31 \( 1 + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-225 - 389. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 6.89e4T^{2} \)
43 \( 1 - 180T + 7.95e4T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (248. - 430. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-370 + 640. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 978.T + 3.57e5T^{2} \)
73 \( 1 + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-692 - 1.19e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 5.71e5T^{2} \)
89 \( 1 + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 9.12e5T^{2} \)
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

−10.83645898411808547505390210367, −9.730234750973137573944729389273, −9.355661018944442646841444153372, −8.125396073160923745933368131661, −7.09219819162166368940436969345, −5.97456635154081055275876411196, −4.92321189606286189974128164240, −3.44311651880787863689911215572, −2.31005036587521560682388428959, −1.12173978266417370166712426020, 0.45574886040584609680642970686, 2.50215889982708529360717775313, 3.67758177154065858239470615053, 5.20773169922782504110776139700, 6.15816486978293473419801527833, 7.09592392411988801053508046220, 7.87181341337110725084601735378, 8.791793130940052569445673180192, 9.411119386956429249456589885756, 10.78165469286358748721310133514

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