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Properties

Label 2-448-112.69-c2-0-20
Degree $2$
Conductor $448$
Sign $0.459 + 0.888i$
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  + (0.746 + 0.746i)3-s + (0.822 − 0.822i)5-s + (6.88 − 1.28i)7-s − 7.88i·9-s + (−2.57 − 2.57i)11-s + (−14.9 − 14.9i)13-s + 1.22·15-s − 20.7i·17-s + (10.0 + 10.0i)19-s + (6.09 + 4.17i)21-s + 19.5i·23-s + 23.6i·25-s + (12.6 − 12.6i)27-s + (7.24 − 7.24i)29-s − 45.5i·31-s + ⋯
L(s)  = 1  + (0.248 + 0.248i)3-s + (0.164 − 0.164i)5-s + (0.982 − 0.183i)7-s − 0.876i·9-s + (−0.234 − 0.234i)11-s + (−1.15 − 1.15i)13-s + 0.0818·15-s − 1.21i·17-s + (0.531 + 0.531i)19-s + (0.290 + 0.198i)21-s + 0.851i·23-s + 0.945i·25-s + (0.466 − 0.466i)27-s + (0.249 − 0.249i)29-s − 1.46i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.459 + 0.888i$
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: \((2,\ 448,\ (\ :1),\ 0.459 + 0.888i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.883850936\)
\(L(\frac12)\) \(\approx\) \(1.883850936\)
\(L(2)\) 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 \( 1 + (-6.88 + 1.28i)T \)
good3 \( 1 + (-0.746 - 0.746i)T + 9iT^{2} \)
5 \( 1 + (-0.822 + 0.822i)T - 25iT^{2} \)
11 \( 1 + (2.57 + 2.57i)T + 121iT^{2} \)
13 \( 1 + (14.9 + 14.9i)T + 169iT^{2} \)
17 \( 1 + 20.7iT - 289T^{2} \)
19 \( 1 + (-10.0 - 10.0i)T + 361iT^{2} \)
23 \( 1 - 19.5iT - 529T^{2} \)
29 \( 1 + (-7.24 + 7.24i)T - 841iT^{2} \)
31 \( 1 + 45.5iT - 961T^{2} \)
37 \( 1 + (34.5 + 34.5i)T + 1.36e3iT^{2} \)
41 \( 1 - 61.5T + 1.68e3T^{2} \)
43 \( 1 + (-16.6 - 16.6i)T + 1.84e3iT^{2} \)
47 \( 1 + 55.0iT - 2.20e3T^{2} \)
53 \( 1 + (-38.2 - 38.2i)T + 2.80e3iT^{2} \)
59 \( 1 + (32.2 - 32.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (32.0 + 32.0i)T + 3.72e3iT^{2} \)
67 \( 1 + (42.1 - 42.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 15.7iT - 5.04e3T^{2} \)
73 \( 1 - 65.7T + 5.32e3T^{2} \)
79 \( 1 - 96.5T + 6.24e3T^{2} \)
83 \( 1 + (-31.6 - 31.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 89.3T + 7.92e3T^{2} \)
97 \( 1 - 47.6iT - 9.40e3T^{2} \)
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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

−10.69056390283002536393666998977, −9.692737143356332119362823769078, −9.117258510932213615887618627092, −7.81090875156561686056362546492, −7.36967029312556339719785345048, −5.73703612079772362444373213781, −5.03806174225260345420894316180, −3.76102965978525587659031670865, −2.53424495470523931022008513366, −0.78480739797527514738195153880, 1.71666347110927652836562161401, 2.61174811043727816508699360277, 4.46305388221486040142095791999, 5.09140669470199778557614677413, 6.51536471955353581037981888474, 7.45755991411861509447840904230, 8.262676499849553242460204215408, 9.102228989550459721395398619389, 10.34012704775345800504835573656, 10.89211680559486756982488594401

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