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Properties

Label 2-448-28.3-c1-0-4
Degree $2$
Conductor $448$
Sign $0.895 - 0.444i$
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  + (0.5 − 0.866i)3-s + (−1.5 + 0.866i)5-s + (2 + 1.73i)7-s + (1 + 1.73i)9-s + (1.5 + 0.866i)11-s + 1.73i·15-s + (−4.5 − 2.59i)17-s + (3.5 + 6.06i)19-s + (2.5 − 0.866i)21-s + (7.5 − 4.33i)23-s + (−1 + 1.73i)25-s + 5·27-s + 6·29-s + (−2.5 + 4.33i)31-s + (1.5 − 0.866i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.670 + 0.387i)5-s + (0.755 + 0.654i)7-s + (0.333 + 0.577i)9-s + (0.452 + 0.261i)11-s + 0.447i·15-s + (−1.09 − 0.630i)17-s + (0.802 + 1.39i)19-s + (0.545 − 0.188i)21-s + (1.56 − 0.902i)23-s + (−0.200 + 0.346i)25-s + 0.962·27-s + 1.11·29-s + (−0.449 + 0.777i)31-s + (0.261 − 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48762 + 0.348492i\)
\(L(\frac12)\) \(\approx\) \(1.48762 + 0.348492i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.5 + 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.5 + 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.5 - 4.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.46iT - 71T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.5 - 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (10.5 - 6.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.19324079712049036509797958602, −10.48035298536420936514595817813, −9.115964111464866245583053545328, −8.390638836177386874955471337784, −7.43874436308053135156227671026, −6.83895733298257019307095787038, −5.35586227041837510054439036943, −4.37956877878679899619220354721, −2.93473144650272399592128102412, −1.66711781056939195470525437197, 1.09531638981840601788368977620, 3.16721748703777296941477063233, 4.28851592292334302512677377763, 4.85175543567999249403828963436, 6.52861183224446142689591277630, 7.39249821190401119780341503246, 8.472037503251095161179955563837, 9.118184707601892226556638047640, 10.08823741864753979791064582967, 11.28239683393158912996302462933

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