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Properties

Label 2-448-112.13-c2-0-12
Degree $2$
Conductor $448$
Sign $-0.631 - 0.775i$
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  + (−1.11 + 1.11i)3-s + (5.47 + 5.47i)5-s + (3.11 + 6.26i)7-s + 6.53i·9-s + (−2.57 + 2.57i)11-s + (−11.1 + 11.1i)13-s − 12.1·15-s − 24.6i·17-s + (10.9 − 10.9i)19-s + (−10.4 − 3.49i)21-s − 10.3i·23-s + 34.9i·25-s + (−17.2 − 17.2i)27-s + (24.5 + 24.5i)29-s − 14.5i·31-s + ⋯
L(s)  = 1  + (−0.370 + 0.370i)3-s + (1.09 + 1.09i)5-s + (0.445 + 0.895i)7-s + 0.725i·9-s + (−0.233 + 0.233i)11-s + (−0.858 + 0.858i)13-s − 0.811·15-s − 1.44i·17-s + (0.578 − 0.578i)19-s + (−0.496 − 0.166i)21-s − 0.451i·23-s + 1.39i·25-s + (−0.639 − 0.639i)27-s + (0.846 + 0.846i)29-s − 0.468i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.631 - 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.631 - 0.775i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.631 - 0.775i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.663119142\)
\(L(\frac12)\) \(\approx\) \(1.663119142\)
\(L(2)\) 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 \( 1 + (-3.11 - 6.26i)T \)
good3 \( 1 + (1.11 - 1.11i)T - 9iT^{2} \)
5 \( 1 + (-5.47 - 5.47i)T + 25iT^{2} \)
11 \( 1 + (2.57 - 2.57i)T - 121iT^{2} \)
13 \( 1 + (11.1 - 11.1i)T - 169iT^{2} \)
17 \( 1 + 24.6iT - 289T^{2} \)
19 \( 1 + (-10.9 + 10.9i)T - 361iT^{2} \)
23 \( 1 + 10.3iT - 529T^{2} \)
29 \( 1 + (-24.5 - 24.5i)T + 841iT^{2} \)
31 \( 1 + 14.5iT - 961T^{2} \)
37 \( 1 + (2.55 - 2.55i)T - 1.36e3iT^{2} \)
41 \( 1 + 48.3T + 1.68e3T^{2} \)
43 \( 1 + (46.0 - 46.0i)T - 1.84e3iT^{2} \)
47 \( 1 + 19.3iT - 2.20e3T^{2} \)
53 \( 1 + (-8.08 + 8.08i)T - 2.80e3iT^{2} \)
59 \( 1 + (-61.0 - 61.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (-75.6 + 75.6i)T - 3.72e3iT^{2} \)
67 \( 1 + (-81.1 - 81.1i)T + 4.48e3iT^{2} \)
71 \( 1 - 9.46iT - 5.04e3T^{2} \)
73 \( 1 - 36.7T + 5.32e3T^{2} \)
79 \( 1 + 55.0T + 6.24e3T^{2} \)
83 \( 1 + (42.2 - 42.2i)T - 6.88e3iT^{2} \)
89 \( 1 + 8.11T + 7.92e3T^{2} \)
97 \( 1 + 141. iT - 9.40e3T^{2} \)
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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

−11.34590565929643624606094236116, −10.09799793137765370581105643199, −9.771728470477498546712883920709, −8.644298122401995447729083834779, −7.29274436249106827265624793533, −6.57471519866942664124237404634, −5.30293604769013222072692218074, −4.86711532966485334469251951266, −2.79539582990017617743372327585, −2.11944368147402317737647647436, 0.72659601980389188377187737269, 1.76578030555955628916013039039, 3.64924484804268999252973567392, 5.01316801316159838175293436464, 5.70800582783671591986134239882, 6.71120817875195717404111840492, 7.903366554022321412650912354182, 8.680313256884968410516579851257, 9.950306023648404753111739888392, 10.23661862641318179644149857334

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