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Properties

Label 2-448-8.3-c2-0-0
Degree $2$
Conductor $448$
Sign $-0.707 - 0.707i$
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  − 1.64·3-s − 7.64i·5-s + 2.64i·7-s − 6.29·9-s − 3.29·11-s − 4.35i·13-s + 12.5i·15-s − 12.5·17-s + 10.3·19-s − 4.35i·21-s + 19.1i·23-s − 33.4·25-s + 25.1·27-s + 52.4i·29-s + 17.1i·31-s + ⋯
L(s)  = 1  − 0.548·3-s − 1.52i·5-s + 0.377i·7-s − 0.699·9-s − 0.299·11-s − 0.334i·13-s + 0.838i·15-s − 0.740·17-s + 0.544·19-s − 0.207i·21-s + 0.833i·23-s − 1.33·25-s + 0.932·27-s + 1.80i·29-s + 0.553i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.07088273017\)
\(L(\frac12)\) \(\approx\) \(0.07088273017\)
\(L(2)\) 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 \( 1 - 2.64iT \)
good3 \( 1 + 1.64T + 9T^{2} \)
5 \( 1 + 7.64iT - 25T^{2} \)
11 \( 1 + 3.29T + 121T^{2} \)
13 \( 1 + 4.35iT - 169T^{2} \)
17 \( 1 + 12.5T + 289T^{2} \)
19 \( 1 - 10.3T + 361T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 - 52.4iT - 841T^{2} \)
31 \( 1 - 17.1iT - 961T^{2} \)
37 \( 1 + 28.4iT - 1.36e3T^{2} \)
41 \( 1 + 50.9T + 1.68e3T^{2} \)
43 \( 1 + 75.2T + 1.84e3T^{2} \)
47 \( 1 - 74.3iT - 2.20e3T^{2} \)
53 \( 1 + 8.91iT - 2.80e3T^{2} \)
59 \( 1 + 45.1T + 3.48e3T^{2} \)
61 \( 1 + 86.4iT - 3.72e3T^{2} \)
67 \( 1 + 113.T + 4.48e3T^{2} \)
71 \( 1 - 67.7iT - 5.04e3T^{2} \)
73 \( 1 + 57.7T + 5.32e3T^{2} \)
79 \( 1 + 62.5iT - 6.24e3T^{2} \)
83 \( 1 - 119.T + 6.88e3T^{2} \)
89 \( 1 - 51.6T + 7.92e3T^{2} \)
97 \( 1 - 164.T + 9.40e3T^{2} \)
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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

−11.38688745371069973774186123136, −10.39644974555964133346683331193, −9.139803754934916966276522306817, −8.725810883977364587295684564949, −7.71286561706217971843946050677, −6.34200884050919643171643512388, −5.23818991390434167561466738830, −4.93160817499285240396037159932, −3.25235682310937367856585079331, −1.48525782396152377015418846692, 0.03125318988394030941208463789, 2.34660790977119378647570770917, 3.39941815012228020057759194399, 4.75613138391992734603460371072, 6.09030820967896908171162989398, 6.65162873035981085985776966289, 7.61032932117946544047278761567, 8.684603664796031107553788665887, 10.07127189283837104639444693794, 10.50965487151575271763665788687

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