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Properties

Label 2-448-7.3-c2-0-5
Degree $2$
Conductor $448$
Sign $-0.749 - 0.661i$
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  + (3.62 + 2.09i)3-s + (−2.74 + 1.58i)5-s + (−2.24 + 6.63i)7-s + (4.24 + 7.34i)9-s + (−6.62 + 11.4i)11-s − 5.49i·13-s − 13.2·15-s + (−11.7 − 6.77i)17-s + (0.621 − 0.358i)19-s + (−21.9 + 19.3i)21-s + (1.13 + 1.96i)23-s + (−7.48 + 12.9i)25-s − 2.15i·27-s − 20.4·29-s + (21.3 + 12.3i)31-s + ⋯
L(s)  = 1  + (1.20 + 0.696i)3-s + (−0.548 + 0.316i)5-s + (−0.320 + 0.947i)7-s + (0.471 + 0.816i)9-s + (−0.601 + 1.04i)11-s − 0.422i·13-s − 0.882·15-s + (−0.690 − 0.398i)17-s + (0.0327 − 0.0188i)19-s + (−1.04 + 0.920i)21-s + (0.0493 + 0.0855i)23-s + (−0.299 + 0.518i)25-s − 0.0797i·27-s − 0.706·29-s + (0.687 + 0.397i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.749 - 0.661i$
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: \((2,\ 448,\ (\ :1),\ -0.749 - 0.661i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.698331864\)
\(L(\frac12)\) \(\approx\) \(1.698331864\)
\(L(2)\) 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 \( 1 + (2.24 - 6.63i)T \)
good3 \( 1 + (-3.62 - 2.09i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.74 - 1.58i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.62 - 11.4i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 5.49iT - 169T^{2} \)
17 \( 1 + (11.7 + 6.77i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-0.621 + 0.358i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-1.13 - 1.96i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 20.4T + 841T^{2} \)
31 \( 1 + (-21.3 - 12.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-32.4 - 56.2i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 21.0iT - 1.68e3T^{2} \)
43 \( 1 + 6.48T + 1.84e3T^{2} \)
47 \( 1 + (-41.3 + 23.8i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-11.0 + 19.0i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-72.5 - 41.8i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (57.3 - 33.1i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-46.3 + 80.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 48.4T + 5.04e3T^{2} \)
73 \( 1 + (-113. - 65.4i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-38.1 - 66.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 107. iT - 6.88e3T^{2} \)
89 \( 1 + (145. - 83.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 25.5iT - 9.40e3T^{2} \)
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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

−11.17141808695311884689802404002, −9.993284898723072282001260494810, −9.500150553213333548273513028856, −8.539283544272564776619581434937, −7.83308971422588216265189749430, −6.77261337945866564665106813965, −5.31670947961665835572101300829, −4.24344168368033718221530382200, −3.11076421535844472803662783485, −2.33750253493311133591020494579, 0.58511038728104594873112412585, 2.23471666449860843274262726539, 3.45495987775247627445337458273, 4.33299449951803508645747914633, 5.99887961843515260428336146389, 7.15229661214220913369188491109, 7.85720087905187474464567979434, 8.538753242584418817733170829667, 9.380472864669899666180128300520, 10.59258469257233519612098460894

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