↑

Properties

Label 2-448-28.3-c3-0-7
Degree $2$
Conductor $448$
Sign $-0.782 - 0.622i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
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  + (−3.44 + 5.96i)3-s + (4.17 − 2.41i)5-s + (5.03 − 17.8i)7-s + (−10.1 − 17.6i)9-s + (−36.6 − 21.1i)11-s − 3.39i·13-s + 33.1i·15-s + (101. + 58.6i)17-s + (45.5 + 78.9i)19-s + (88.8 + 91.3i)21-s + (−147. + 85.3i)23-s + (−50.8 + 88.1i)25-s − 45.6·27-s + 131.·29-s + (−5.70 + 9.87i)31-s + ⋯
L(s)  = 1  + (−0.662 + 1.14i)3-s + (0.373 − 0.215i)5-s + (0.272 − 0.962i)7-s + (−0.377 − 0.653i)9-s + (−1.00 − 0.579i)11-s − 0.0724i·13-s + 0.571i·15-s + (1.45 + 0.837i)17-s + (0.550 + 0.953i)19-s + (0.923 + 0.949i)21-s + (−1.33 + 0.773i)23-s + (−0.406 + 0.704i)25-s − 0.325·27-s + 0.841·29-s + (−0.0330 + 0.0572i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.782 - 0.622i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.782 - 0.622i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.9786045858\)
\(L(\frac12)\) \(\approx\) \(0.9786045858\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-5.03 + 17.8i)T \)
good3 \( 1 + (3.44 - 5.96i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-4.17 + 2.41i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (36.6 + 21.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 3.39iT - 2.19e3T^{2} \)
17 \( 1 + (-101. - 58.6i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-45.5 - 78.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (147. - 85.3i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 131.T + 2.43e4T^{2} \)
31 \( 1 + (5.70 - 9.87i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (59.2 + 102. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 109. iT - 6.89e4T^{2} \)
43 \( 1 - 82.5iT - 7.95e4T^{2} \)
47 \( 1 + (-36.7 - 63.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (87.2 - 151. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (166. - 288. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (472. - 272. i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-516. - 298. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 384. iT - 3.57e5T^{2} \)
73 \( 1 + (-187. - 108. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (868. - 501. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 459.T + 5.71e5T^{2} \)
89 \( 1 + (771. - 445. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 282. iT - 9.12e5T^{2} \)
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

−10.79195977851275265506483674725, −10.11169962021804517780221553925, −9.761777665660477978240449785424, −8.207818776719067596063979019764, −7.55776280547090130494402977736, −5.82304174672510778235950034670, −5.47042758481426086999356899397, −4.24507241413826529924750124010, −3.38471534333999332130629190495, −1.34124403963672520815440318769, 0.35651491020633933755052363947, 1.87647071546003143583595327901, 2.81988505114930180255254525144, 4.88510180976784268536096999900, 5.68518796841016017940604368171, 6.52388427225962262935285311724, 7.52015552616444286349616999205, 8.220483386025706468932690799666, 9.554780069420556038056548767789, 10.33115339846144778421754963233

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