↑

Properties

Label 2-448-112.19-c1-0-3
Degree $2$
Conductor $448$
Sign $0.685 - 0.727i$
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  + (0.282 − 1.05i)3-s + (−1.39 + 0.374i)5-s + (0.298 + 2.62i)7-s + (1.56 + 0.904i)9-s + (−1.42 + 5.30i)11-s + (−1.84 + 1.84i)13-s + 1.58i·15-s + (5.69 − 3.28i)17-s + (3.24 − 0.869i)19-s + (2.85 + 0.427i)21-s + (0.256 − 0.443i)23-s + (−2.51 + 1.45i)25-s + (3.71 − 3.71i)27-s + (−2.30 − 2.30i)29-s + (3.79 + 6.57i)31-s + ⋯
L(s)  = 1  + (0.163 − 0.608i)3-s + (−0.625 + 0.167i)5-s + (0.112 + 0.993i)7-s + (0.522 + 0.301i)9-s + (−0.428 + 1.60i)11-s + (−0.511 + 0.511i)13-s + 0.408i·15-s + (1.38 − 0.797i)17-s + (0.744 − 0.199i)19-s + (0.623 + 0.0933i)21-s + (0.0534 − 0.0925i)23-s + (−0.502 + 0.290i)25-s + (0.714 − 0.714i)27-s + (−0.427 − 0.427i)29-s + (0.681 + 1.18i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.685 - 0.727i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.685 - 0.727i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19406 + 0.515655i\)
\(L(\frac12)\) \(\approx\) \(1.19406 + 0.515655i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.298 - 2.62i)T \)
good3 \( 1 + (-0.282 + 1.05i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.39 - 0.374i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.42 - 5.30i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.84 - 1.84i)T - 13iT^{2} \)
17 \( 1 + (-5.69 + 3.28i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.24 + 0.869i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.256 + 0.443i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.30 + 2.30i)T + 29iT^{2} \)
31 \( 1 + (-3.79 - 6.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.07 - 4.01i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.453T + 41T^{2} \)
43 \( 1 + (3.40 + 3.40i)T + 43iT^{2} \)
47 \( 1 + (-1.71 + 2.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.37 + 0.367i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (6.57 + 1.76i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.30 - 4.88i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.19 - 1.66i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 6.44T + 71T^{2} \)
73 \( 1 + (-7.43 - 12.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.33 - 1.92i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.44 + 2.44i)T + 83iT^{2} \)
89 \( 1 + (-3.81 + 6.60i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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

−11.57489026042911801257707707311, −10.07527529038842233794724035836, −9.584556258479088157975917301144, −8.254518888857002056754680893096, −7.44063027609835211551391250596, −6.95734247541777774976186716385, −5.38241921663366662152414896089, −4.55217189281041832839003565589, −2.93078011990085584841537008516, −1.77428451328871152862753588353, 0.868285025987681711364907859869, 3.30841358175798184323191573459, 3.87721989204976209782694796675, 5.10182323271931530127073498699, 6.20748820321096169728539575664, 7.75751597803320447587726585823, 7.920743605291331257445741108062, 9.325436183154578703980089536405, 10.18454165629130465246821163122, 10.80253285902103599620616159203

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