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Properties

Label 2-448-112.69-c2-0-2
Degree $2$
Conductor $448$
Sign $0.344 - 0.938i$
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  + (−1.79 − 1.79i)3-s + (−3.85 + 3.85i)5-s + (6.95 + 0.829i)7-s − 2.54i·9-s + (−14.1 − 14.1i)11-s + (5.62 + 5.62i)13-s + 13.8·15-s + 23.7i·17-s + (6.44 + 6.44i)19-s + (−10.9 − 13.9i)21-s + 15.1i·23-s − 4.68i·25-s + (−20.7 + 20.7i)27-s + (−10.0 + 10.0i)29-s − 22.6i·31-s + ⋯
L(s)  = 1  + (−0.598 − 0.598i)3-s + (−0.770 + 0.770i)5-s + (0.992 + 0.118i)7-s − 0.282i·9-s + (−1.28 − 1.28i)11-s + (0.433 + 0.433i)13-s + 0.922·15-s + 1.39i·17-s + (0.339 + 0.339i)19-s + (−0.523 − 0.665i)21-s + 0.657i·23-s − 0.187i·25-s + (−0.768 + 0.768i)27-s + (−0.346 + 0.346i)29-s − 0.731i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ 0.344 - 0.938i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.8973293643\)
\(L(\frac12)\) \(\approx\) \(0.8973293643\)
\(L(2)\) 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 \( 1 + (-6.95 - 0.829i)T \)
good3 \( 1 + (1.79 + 1.79i)T + 9iT^{2} \)
5 \( 1 + (3.85 - 3.85i)T - 25iT^{2} \)
11 \( 1 + (14.1 + 14.1i)T + 121iT^{2} \)
13 \( 1 + (-5.62 - 5.62i)T + 169iT^{2} \)
17 \( 1 - 23.7iT - 289T^{2} \)
19 \( 1 + (-6.44 - 6.44i)T + 361iT^{2} \)
23 \( 1 - 15.1iT - 529T^{2} \)
29 \( 1 + (10.0 - 10.0i)T - 841iT^{2} \)
31 \( 1 + 22.6iT - 961T^{2} \)
37 \( 1 + (-31.9 - 31.9i)T + 1.36e3iT^{2} \)
41 \( 1 - 49.4T + 1.68e3T^{2} \)
43 \( 1 + (-34.1 - 34.1i)T + 1.84e3iT^{2} \)
47 \( 1 - 82.6iT - 2.20e3T^{2} \)
53 \( 1 + (-11.1 - 11.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (70.8 - 70.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (24.6 + 24.6i)T + 3.72e3iT^{2} \)
67 \( 1 + (-7.08 + 7.08i)T - 4.48e3iT^{2} \)
71 \( 1 + 43.8iT - 5.04e3T^{2} \)
73 \( 1 + 63.2T + 5.32e3T^{2} \)
79 \( 1 + 36.2T + 6.24e3T^{2} \)
83 \( 1 + (-22.7 - 22.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 2.50T + 7.92e3T^{2} \)
97 \( 1 - 135. iT - 9.40e3T^{2} \)
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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

−11.09234229913420672451861162955, −10.70484032136278593397942340631, −9.129138364164695481938401459588, −7.87874407195174558747704050384, −7.68478048599498154198598268752, −6.21676310921532897749534803155, −5.68277928169535212661341587970, −4.15336278227545024630650341973, −3.00079039127602020410438541279, −1.26682378904803981409856031715, 0.45139530053404478093919052407, 2.37301713933502503173542719273, 4.26975546859102997025375711370, 4.87324323105758574961872925529, 5.47572944596862726019621196766, 7.36765083246563887899546243192, 7.82840992442922341236876716011, 8.866403708071242047139297415369, 10.01977128893506260147909633696, 10.77724753331113884813044044146

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