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Properties

Label 2-448-7.5-c2-0-24
Degree $2$
Conductor $448$
Sign $-0.314 + 0.949i$
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  + (−0.438 + 0.253i)3-s + (4.59 + 2.65i)5-s + (−5.27 − 4.59i)7-s + (−4.37 + 7.57i)9-s + (−8.54 − 14.8i)11-s − 21.4i·13-s − 2.68·15-s + (−20.7 + 12.0i)17-s + (10.5 + 6.11i)19-s + (3.48 + 0.680i)21-s + (20.1 − 34.8i)23-s + (1.55 + 2.69i)25-s − 8.99i·27-s + 26.0·29-s + (−21.8 + 12.6i)31-s + ⋯
L(s)  = 1  + (−0.146 + 0.0844i)3-s + (0.918 + 0.530i)5-s + (−0.753 − 0.656i)7-s + (−0.485 + 0.841i)9-s + (−0.776 − 1.34i)11-s − 1.65i·13-s − 0.179·15-s + (−1.22 + 0.706i)17-s + (0.557 + 0.321i)19-s + (0.165 + 0.0324i)21-s + (0.875 − 1.51i)23-s + (0.0623 + 0.107i)25-s − 0.333i·27-s + 0.899·29-s + (−0.705 + 0.407i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.314 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.314 + 0.949i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.314 + 0.949i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9637992031\)
\(L(\frac12)\) \(\approx\) \(0.9637992031\)
\(L(2)\) 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 \( 1 + (5.27 + 4.59i)T \)
good3 \( 1 + (0.438 - 0.253i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-4.59 - 2.65i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (8.54 + 14.8i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + (20.7 - 12.0i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-10.5 - 6.11i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-20.1 + 34.8i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 26.0T + 841T^{2} \)
31 \( 1 + (21.8 - 12.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-6.48 + 11.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 33.8iT - 1.68e3T^{2} \)
43 \( 1 + 29.9T + 1.84e3T^{2} \)
47 \( 1 + (48.2 + 27.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (4.36 + 7.55i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-43.2 + 24.9i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (3.40 + 1.96i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (52.9 + 91.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 35.0T + 5.04e3T^{2} \)
73 \( 1 + (-40.3 + 23.3i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (43.4 - 75.3i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 64.0iT - 6.88e3T^{2} \)
89 \( 1 + (-37.2 - 21.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 28.7iT - 9.40e3T^{2} \)
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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

−10.62095706624727271287843421962, −10.11183580501452752447953811154, −8.697526634621363704394325418678, −7.984700928989802508584515768399, −6.67759813811856150088061275579, −5.90915742476095043200210155285, −5.05774005678415825629058893575, −3.32711250226081262572379349123, −2.51666639328654133370944388851, −0.38334458258137809137517049871, 1.73567636149457240657806690980, 2.89900031180577925372540290922, 4.57516571903674729903845211599, 5.46415408791127061500447050205, 6.51743576714067810265158824692, 7.19185387612281950135763024824, 8.892103670473456411718792490731, 9.374228298685148223483069126368, 9.844959807915324626263533744222, 11.45080590722750276112104994922

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