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Properties

Label 2-448-7.3-c2-0-7
Degree $2$
Conductor $448$
Sign $0.232 + 0.972i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.97 − 2.86i)3-s + (−5.45 + 3.15i)5-s + (−5.64 + 4.13i)7-s + (11.9 + 20.7i)9-s + (2.70 − 4.68i)11-s + 15.9i·13-s + 36.1·15-s + (−17.7 − 10.2i)17-s + (−11.7 + 6.79i)19-s + (39.9 − 4.32i)21-s + (−2.35 − 4.07i)23-s + (7.37 − 12.7i)25-s − 85.7i·27-s − 1.76·29-s + (−11.9 − 6.87i)31-s + ⋯
L(s)  = 1  + (−1.65 − 0.956i)3-s + (−1.09 + 0.630i)5-s + (−0.807 + 0.590i)7-s + (1.33 + 2.30i)9-s + (0.245 − 0.426i)11-s + 1.22i·13-s + 2.41·15-s + (−1.04 − 0.602i)17-s + (−0.619 + 0.357i)19-s + (1.90 − 0.206i)21-s + (−0.102 − 0.177i)23-s + (0.294 − 0.510i)25-s − 3.17i·27-s − 0.0608·29-s + (−0.383 − 0.221i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.232 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.232 + 0.972i$
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: \((2,\ 448,\ (\ :1),\ 0.232 + 0.972i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2685583469\)
\(L(\frac12)\) \(\approx\) \(0.2685583469\)
\(L(2)\) 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 \( 1 + (5.64 - 4.13i)T \)
good3 \( 1 + (4.97 + 2.86i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (5.45 - 3.15i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.70 + 4.68i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 15.9iT - 169T^{2} \)
17 \( 1 + (17.7 + 10.2i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (11.7 - 6.79i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (2.35 + 4.07i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 1.76T + 841T^{2} \)
31 \( 1 + (11.9 + 6.87i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-5.23 - 9.06i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 11.2iT - 1.68e3T^{2} \)
43 \( 1 + 49.1T + 1.84e3T^{2} \)
47 \( 1 + (9.02 - 5.20i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-16.0 + 27.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-57.2 - 33.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-27.6 + 15.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-49.2 + 85.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 61.7T + 5.04e3T^{2} \)
73 \( 1 + (-15.6 - 9.02i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (15.0 + 25.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 63.4iT - 6.88e3T^{2} \)
89 \( 1 + (-119. + 68.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26619114875910181347730885277, −10.09998448481195989771223957596, −8.770949487889241863787018690520, −7.57036103750334478015856616610, −6.62215548596226676444850526819, −6.42591700164402368898254513156, −5.08989396668826735976140136385, −3.89220624277193244300805467474, −2.14289444919463123856866535155, −0.24991488335783971947874569238, 0.65056268528292452668766726523, 3.66628377930605556412242473627, 4.30005788369405267470424589651, 5.20513229060341107611164481627, 6.30596900804252123692451288312, 7.11787947685007050107045493938, 8.451169427867936543531793985570, 9.582389521469057297547519004629, 10.38087401692941224352845523981, 11.02280362434208593830763517523

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