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Properties

Label 2-448-28.23-c2-0-29
Degree $2$
Conductor $448$
Sign $-0.311 + 0.950i$
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  + (4.33 − 2.5i)3-s + (4.5 − 7.79i)5-s + (−6.92 + i)7-s + (8.00 − 13.8i)9-s + (2.59 − 1.5i)11-s − 16·13-s − 45.0i·15-s + (3.5 + 6.06i)17-s + (9.52 + 5.5i)19-s + (−27.5 + 21.6i)21-s + (16.4 + 9.5i)23-s + (−28 − 48.4i)25-s − 35.0i·27-s + 32·29-s + (−9.52 + 5.5i)31-s + ⋯
L(s)  = 1  + (1.44 − 0.833i)3-s + (0.900 − 1.55i)5-s + (−0.989 + 0.142i)7-s + (0.888 − 1.53i)9-s + (0.236 − 0.136i)11-s − 1.23·13-s − 3.00i·15-s + (0.205 + 0.356i)17-s + (0.501 + 0.289i)19-s + (−1.30 + 1.03i)21-s + (0.715 + 0.413i)23-s + (−1.12 − 1.93i)25-s − 1.29i·27-s + 1.10·29-s + (−0.307 + 0.177i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.311 + 0.950i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.311 + 0.950i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.848752494\)
\(L(\frac12)\) \(\approx\) \(2.848752494\)
\(L(2)\) 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 \( 1 + (6.92 - i)T \)
good3 \( 1 + (-4.33 + 2.5i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.5 + 7.79i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.59 + 1.5i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-9.52 - 5.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-16.4 - 9.5i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 32T + 841T^{2} \)
31 \( 1 + (9.52 - 5.5i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 40T + 1.68e3T^{2} \)
43 \( 1 - 40iT - 1.84e3T^{2} \)
47 \( 1 + (-73.6 - 42.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-3.5 - 6.06i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-45.8 + 26.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-39.5 + 68.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (9.52 - 5.5i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48iT - 5.04e3T^{2} \)
73 \( 1 + (71.5 + 123. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-30.3 - 17.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 8iT - 6.88e3T^{2} \)
89 \( 1 + (-48.5 + 84.0i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 88T + 9.40e3T^{2} \)
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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

−10.07093394799684994249570757178, −9.420048527153302206347843689591, −8.914542144421389664757639055992, −8.055147340055703859277886074978, −7.05946966802863971181015050224, −5.96681358950092257998367867366, −4.79510165968560233979055911569, −3.30532309726928598219755974303, −2.20156947223036813898365000328, −1.03791526467424555354298566892, 2.48559893247186060354692811743, 2.87427947598787424818679799056, 3.92589253053704072683227497190, 5.40578599409973778717773110376, 6.87099201473158836969679817910, 7.25049481133839713907164713491, 8.734987879829572176730892588731, 9.606634025885570851690001310456, 10.04616259878948546009593871740, 10.60483122704516630558517448815

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