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Properties

Label 2-448-7.4-c1-0-1
Degree $2$
Conductor $448$
Sign $-0.0725 - 0.997i$
Analytic cond. $3.57729$
Root an. cond. $1.89137$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 + 0.358i)3-s + (−0.914 − 1.58i)5-s + (−1 + 2.44i)7-s + (1.41 + 2.44i)9-s + (−1.20 + 2.09i)11-s − 2.82·13-s + 0.757·15-s + (0.0857 − 0.148i)17-s + (3.20 + 5.55i)19-s + (−0.671 − 0.866i)21-s + (2.62 + 4.54i)23-s + (0.828 − 1.43i)25-s − 2.41·27-s + 2.82·29-s + (−2.79 + 4.83i)31-s + ⋯
L(s)  = 1  + (−0.119 + 0.207i)3-s + (−0.408 − 0.708i)5-s + (−0.377 + 0.925i)7-s + (0.471 + 0.816i)9-s + (−0.363 + 0.630i)11-s − 0.784·13-s + 0.195·15-s + (0.0208 − 0.0360i)17-s + (0.735 + 1.27i)19-s + (−0.146 − 0.188i)21-s + (0.546 + 0.946i)23-s + (0.165 − 0.286i)25-s − 0.464·27-s + 0.525·29-s + (−0.501 + 0.868i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.0725 - 0.997i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ -0.0725 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.659192 + 0.708894i\)
\(L(\frac12)\) \(\approx\) \(0.659192 + 0.708894i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2.44i)T \)
good3 \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.914 + 1.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.20 - 2.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 + (-0.0857 + 0.148i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.20 - 5.55i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.62 - 4.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (2.79 - 4.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.32 - 7.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.82T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 + (5.20 + 9.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.44 + 9.43i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.32 - 7.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.37 - 2.38i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (-7.32 + 12.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.31T + 83T^{2} \)
89 \( 1 + (-4.5 - 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.17T + 97T^{2} \)
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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

−11.55119768788563491093807703119, −10.13149505297603394641852147224, −9.762464965702838080531567489925, −8.533399731774335095735356787707, −7.82083917459625762814060130512, −6.75195004197372641543140633103, −5.23754296771729787380190483317, −4.90888593548456703371061188053, −3.36722949478731769330955766202, −1.86296701807325539386889680588, 0.61948367379206650898328426888, 2.83168037886774448896074089838, 3.79336242746594523140399416328, 5.02991398503370514751615415304, 6.54120658042557691256455019767, 7.02500977325575792464335640147, 7.87713352570676301230496992873, 9.222516880965004924801455184258, 10.01290893501265768808526386761, 10.95614411977874142871229350556

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