Properties

Label 2-448-7.3-c2-0-23
Degree $2$
Conductor $448$
Sign $0.871 + 0.490i$
Analytic cond. $12.2071$
Root an. cond. $3.49386$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (3.19 + 1.84i)3-s + (2.63 − 1.52i)5-s + (0.812 − 6.95i)7-s + (2.28 + 3.96i)9-s + (1.17 − 2.03i)11-s − 25.3i·13-s + 11.2·15-s + (3.08 + 1.78i)17-s + (14.1 − 8.18i)19-s + (15.4 − 20.6i)21-s + (8.83 + 15.3i)23-s + (−7.85 + 13.6i)25-s − 16.2i·27-s − 36.1·29-s + (6.25 + 3.61i)31-s + ⋯
L(s)  = 1  + (1.06 + 0.614i)3-s + (0.527 − 0.304i)5-s + (0.116 − 0.993i)7-s + (0.254 + 0.440i)9-s + (0.106 − 0.185i)11-s − 1.94i·13-s + 0.748·15-s + (0.181 + 0.104i)17-s + (0.746 − 0.430i)19-s + (0.733 − 0.985i)21-s + (0.384 + 0.665i)23-s + (−0.314 + 0.544i)25-s − 0.603i·27-s − 1.24·29-s + (0.201 + 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.871 + 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.871 + 0.490i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.871 + 0.490i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.691522109\)
\(L(\frac12)\) \(\approx\) \(2.691522109\)
\(L(2)\) 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 \( 1 + (-0.812 + 6.95i)T \)
good3 \( 1 + (-3.19 - 1.84i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-2.63 + 1.52i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 25.3iT - 169T^{2} \)
17 \( 1 + (-3.08 - 1.78i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.1 + 8.18i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-8.83 - 15.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 36.1T + 841T^{2} \)
31 \( 1 + (-6.25 - 3.61i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-18.4 - 31.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 53.7iT - 1.68e3T^{2} \)
43 \( 1 - 51.2T + 1.84e3T^{2} \)
47 \( 1 + (27.1 - 15.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (35.1 - 60.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-81.4 - 47.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-1.89 + 1.09i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-12.4 + 21.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 50.8T + 5.04e3T^{2} \)
73 \( 1 + (68.9 + 39.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-57.5 - 99.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 154. iT - 6.88e3T^{2} \)
89 \( 1 + (-98.7 + 57.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 53.9iT - 9.40e3T^{2} \)
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

−10.55297869175768604167120415279, −9.783271039252336160107352008078, −9.194873416068035517152575115296, −8.034760020852551499406481445042, −7.51285486814083521491362059638, −5.94597478094911139262833050105, −4.93494787183548746210706094716, −3.65784830331499686688340797359, −2.91213920178664782974234006545, −1.07684681945363863512109566371, 1.87074318743060312254976197219, 2.43120477615304986341673176902, 3.85414333836155460471153145404, 5.33179556091908976329173766776, 6.45399794112752892328679503855, 7.30677618523229416685410306789, 8.326958493882966384778372180708, 9.172859924801108087832951536555, 9.638307501979673618266098750181, 11.08923930635096199613704529437

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