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Properties

Label 2-448-7.4-c3-0-6
Degree $2$
Conductor $448$
Sign $-0.857 - 0.514i$
Analytic cond. $26.4328$
Root an. cond. $5.14128$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (3.5 + 6.06i)5-s + (10 + 15.5i)7-s + (13 + 22.5i)9-s + (−17.5 + 30.3i)11-s − 66·13-s + 7·15-s + (−29.5 + 51.0i)17-s + (−68.5 − 118. i)19-s + (18.5 − 0.866i)21-s + (−3.5 − 6.06i)23-s + (38 − 65.8i)25-s + 53·27-s − 106·29-s + (37.5 − 64.9i)31-s + ⋯
L(s)  = 1  + (0.0962 − 0.166i)3-s + (0.313 + 0.542i)5-s + (0.539 + 0.841i)7-s + (0.481 + 0.833i)9-s + (−0.479 + 0.830i)11-s − 1.40·13-s + 0.120·15-s + (−0.420 + 0.728i)17-s + (−0.827 − 1.43i)19-s + (0.192 − 0.00899i)21-s + (−0.0317 − 0.0549i)23-s + (0.303 − 0.526i)25-s + 0.377·27-s − 0.678·29-s + (0.217 − 0.376i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.857 - 0.514i$
Analytic conductor: \(26.4328\)
Root analytic conductor: \(5.14128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :3/2),\ -0.857 - 0.514i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.173395491\)
\(L(\frac12)\) \(\approx\) \(1.173395491\)
\(L(\frac{5}{2})\) 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 \( 1 + (-10 - 15.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + (-3.5 - 6.06i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (17.5 - 30.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 66T + 2.19e3T^{2} \)
17 \( 1 + (29.5 - 51.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (68.5 + 118. i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (3.5 + 6.06i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 106T + 2.43e4T^{2} \)
31 \( 1 + (-37.5 + 64.9i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-5.5 - 9.52i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 498T + 6.89e4T^{2} \)
43 \( 1 - 260T + 7.95e4T^{2} \)
47 \( 1 + (85.5 + 148. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (208.5 - 361. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-8.5 + 14.7i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-25.5 - 44.1i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (219.5 - 380. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 784T + 3.57e5T^{2} \)
73 \( 1 + (147.5 - 255. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (247.5 + 428. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 932T + 5.71e5T^{2} \)
89 \( 1 + (-436.5 - 756. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 290T + 9.12e5T^{2} \)
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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

−10.89018575255869415257165054881, −10.26695291390600637660926704759, −9.294386077537620049015046292801, −8.246776075433042346472371999410, −7.36688100243783916084867516222, −6.53982556163929044772153669574, −5.15998357402487399080045436611, −4.53700202195283974534881579335, −2.49904296925001619361700396718, −2.08528510400374201455994199123, 0.34483778387656788667535753200, 1.73084276041101222266653045108, 3.36200310475720081485516347963, 4.49867657076258261340620555795, 5.35513684061726215272377902286, 6.63923243457239113338874015232, 7.58427613725871253583129842705, 8.504815803251430051804075832059, 9.523617809469930483553996428802, 10.19644400962546850298101090235

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