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Properties

Label 2-448-28.3-c3-0-30
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

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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

\(L(2)\) \(\approx\) \(2.850211640\)
\(L(\frac12)\) \(\approx\) \(2.850211640\)
\(L(\frac{5}{2})\) 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 \( 1 - 282. iT - 9.12e5T^{2} \)
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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

−10.52753457085898610657254536095, −9.334896388655536580931463837888, −8.812669494135185646769705574560, −7.86480153952560603037399526455, −6.85555103450285970125976489606, −6.13027824668936698329548464068, −4.89417187371466157700160127753, −3.26033627589690143831309798271, −2.17032017880016278473201036482, −1.16607347351574228249982888723, 1.11962340652743862020918371647, 3.08321303015698744257885326189, 3.72071940547298640950536185882, 4.75390973820078525320876767338, 6.05050567079704686815194251902, 7.10264216088909246507247368155, 8.229060255652287892795077772355, 9.211651024457236694577954079459, 9.907927049877761182255714359648, 10.42966036296278717466163149272

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