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Properties

Label 2-448-7.3-c2-0-24
Degree $2$
Conductor $448$
Sign $-0.868 + 0.495i$
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  + (−3.45 − 1.99i)3-s + (7.80 − 4.50i)5-s + (−5.54 − 4.26i)7-s + (3.44 + 5.96i)9-s + (8.28 − 14.3i)11-s − 0.446i·13-s − 35.9·15-s + (6.02 + 3.47i)17-s + (11.0 − 6.37i)19-s + (10.6 + 25.7i)21-s + (13.2 + 23.0i)23-s + (28.1 − 48.7i)25-s + 8.43i·27-s − 26.4·29-s + (−21.7 − 12.5i)31-s + ⋯
L(s)  = 1  + (−1.15 − 0.664i)3-s + (1.56 − 0.901i)5-s + (−0.792 − 0.609i)7-s + (0.382 + 0.662i)9-s + (0.752 − 1.30i)11-s − 0.0343i·13-s − 2.39·15-s + (0.354 + 0.204i)17-s + (0.580 − 0.335i)19-s + (0.507 + 1.22i)21-s + (0.577 + 1.00i)23-s + (1.12 − 1.95i)25-s + 0.312i·27-s − 0.912·29-s + (−0.702 − 0.405i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.868 + 0.495i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.868 + 0.495i$
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: \((2,\ 448,\ (\ :1),\ -0.868 + 0.495i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.245111396\)
\(L(\frac12)\) \(\approx\) \(1.245111396\)
\(L(2)\) 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 \( 1 + (5.54 + 4.26i)T \)
good3 \( 1 + (3.45 + 1.99i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-7.80 + 4.50i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-8.28 + 14.3i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 0.446iT - 169T^{2} \)
17 \( 1 + (-6.02 - 3.47i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.0 + 6.37i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-13.2 - 23.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 26.4T + 841T^{2} \)
31 \( 1 + (21.7 + 12.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (31.6 + 54.9i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 0.519iT - 1.68e3T^{2} \)
43 \( 1 - 25.5T + 1.84e3T^{2} \)
47 \( 1 + (59.4 - 34.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (3.58 - 6.20i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (65.3 + 37.7i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (39.8 - 23.0i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-21.4 + 37.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 60.0T + 5.04e3T^{2} \)
73 \( 1 + (40.5 + 23.3i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-27.1 - 47.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 11.4iT - 6.88e3T^{2} \)
89 \( 1 + (-53.1 + 30.7i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 20.3iT - 9.40e3T^{2} \)
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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

−10.65868613110615705647311465512, −9.434129264290814279749130814056, −9.103239019531512950990894478977, −7.48923065523319728664813514623, −6.39985516706761129115117277189, −5.87019705483561249496464719875, −5.18733362825587178561851859274, −3.47742154145094654020476223222, −1.54811571715710086167565174228, −0.62244674490424530008265743535, 1.87242210387729759941084365217, 3.21244969372795396091585202192, 4.83364713113514001417639367902, 5.66556334229741237986131294486, 6.42293265353422376462088189641, 7.06816262282560273970619343776, 9.054119363296108421312539785880, 9.838306226974168798777725686812, 10.14128499800675245977347242339, 11.09540842008331684947108224954

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