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Properties

Label 2-448-112.13-c2-0-20
Degree $2$
Conductor $448$
Sign $0.939 - 0.342i$
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  + (−3.98 + 3.98i)3-s + (2.85 + 2.85i)5-s + (6.99 − 0.139i)7-s − 22.7i·9-s + (7.34 − 7.34i)11-s + (7.74 − 7.74i)13-s − 22.7·15-s − 16.9i·17-s + (9.10 − 9.10i)19-s + (−27.3 + 28.4i)21-s − 38.7i·23-s − 8.66i·25-s + (54.7 + 54.7i)27-s + (14.6 + 14.6i)29-s + 8.79i·31-s + ⋯
L(s)  = 1  + (−1.32 + 1.32i)3-s + (0.571 + 0.571i)5-s + (0.999 − 0.0199i)7-s − 2.52i·9-s + (0.667 − 0.667i)11-s + (0.595 − 0.595i)13-s − 1.51·15-s − 0.994i·17-s + (0.479 − 0.479i)19-s + (−1.30 + 1.35i)21-s − 1.68i·23-s − 0.346i·25-s + (2.02 + 2.02i)27-s + (0.505 + 0.505i)29-s + 0.283i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.389637004\)
\(L(\frac12)\) \(\approx\) \(1.389637004\)
\(L(2)\) 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 \( 1 + (-6.99 + 0.139i)T \)
good3 \( 1 + (3.98 - 3.98i)T - 9iT^{2} \)
5 \( 1 + (-2.85 - 2.85i)T + 25iT^{2} \)
11 \( 1 + (-7.34 + 7.34i)T - 121iT^{2} \)
13 \( 1 + (-7.74 + 7.74i)T - 169iT^{2} \)
17 \( 1 + 16.9iT - 289T^{2} \)
19 \( 1 + (-9.10 + 9.10i)T - 361iT^{2} \)
23 \( 1 + 38.7iT - 529T^{2} \)
29 \( 1 + (-14.6 - 14.6i)T + 841iT^{2} \)
31 \( 1 - 8.79iT - 961T^{2} \)
37 \( 1 + (5.59 - 5.59i)T - 1.36e3iT^{2} \)
41 \( 1 + 0.648T + 1.68e3T^{2} \)
43 \( 1 + (11.6 - 11.6i)T - 1.84e3iT^{2} \)
47 \( 1 - 5.84iT - 2.20e3T^{2} \)
53 \( 1 + (-34.7 + 34.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (4.12 + 4.12i)T + 3.48e3iT^{2} \)
61 \( 1 + (32.6 - 32.6i)T - 3.72e3iT^{2} \)
67 \( 1 + (74.0 + 74.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 24.7iT - 5.04e3T^{2} \)
73 \( 1 + 95.7T + 5.32e3T^{2} \)
79 \( 1 - 43.7T + 6.24e3T^{2} \)
83 \( 1 + (82.9 - 82.9i)T - 6.88e3iT^{2} \)
89 \( 1 - 93.1T + 7.92e3T^{2} \)
97 \( 1 - 116. iT - 9.40e3T^{2} \)
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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

−10.75717726379413822509555004658, −10.42963139728713140604829961917, −9.349697937438695797177075704279, −8.486845508731947763159558904563, −6.82888252559572948191840678386, −6.05460719757710083713503357678, −5.15837969888556596763733695255, −4.37321854958785810329634888144, −3.05139313578685750147213282878, −0.809741605654218621305656955250, 1.39455039934565083793613110229, 1.70186308119696109543087055437, 4.31225996782907137088920982884, 5.44472993866289313438843324346, 5.98812105657881347490808798261, 7.08266646167504829946838212722, 7.83903160660895336706802665395, 8.903849578019190989498751420661, 10.13853916343200174541257943831, 11.20053278010051058609018595396

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