↑

Properties

Label 2-448-7.2-c1-0-11
Degree $2$
Conductor $448$
Sign $0.386 + 0.922i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
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  + (0.5 + 0.866i)3-s + (1.5 − 2.59i)5-s + (−2 − 1.73i)7-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s − 2·13-s + 3·15-s + (−1.5 − 2.59i)17-s + (−0.5 + 0.866i)19-s + (0.499 − 2.59i)21-s + (−1.5 + 2.59i)23-s + (−2 − 3.46i)25-s + 5·27-s + 6·29-s + (3.5 + 6.06i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.670 − 1.16i)5-s + (−0.755 − 0.654i)7-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s − 0.554·13-s + 0.774·15-s + (−0.363 − 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.109 − 0.566i)21-s + (−0.312 + 0.541i)23-s + (−0.400 − 0.692i)25-s + 0.962·27-s + 1.11·29-s + (0.628 + 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22066 - 0.811963i\)
\(L(\frac12)\) \(\approx\) \(1.22066 - 0.811963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−10.60216375342673559217309081292, −9.916742350002390261269684926635, −9.213615508919535885049459952993, −8.511452802147931166586230345484, −7.21244568690470416737831698831, −6.14366164945005922025632076894, −5.06494523167841989402454105010, −4.10174630475570383932116435597, −2.86797232150328807388802996516, −0.907207386151134318322245084356, 2.25511409019656162972402365717, 2.69291512271974295727268605821, 4.46678606196403938125379725520, 5.87222181691419343099747319456, 6.66863239806856136155009035563, 7.42839370953603325488625656554, 8.473733715006804911329519167331, 9.750384490253818090029257775479, 10.19387669536850782414458407309, 11.11029907321283449217431173580

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