Properties

Label 2-448-448.227-c1-0-32
Degree $2$
Conductor $448$
Sign $0.786 - 0.616i$
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  + (−0.425 + 1.34i)2-s + (−1.61 + 1.84i)3-s + (−1.63 − 1.14i)4-s + (−0.00363 + 0.0554i)5-s + (−1.79 − 2.96i)6-s + (2.12 − 1.57i)7-s + (2.24 − 1.71i)8-s + (−0.391 − 2.97i)9-s + (−0.0732 − 0.0285i)10-s + (1.96 − 5.78i)11-s + (4.75 − 1.16i)12-s + (2.24 − 1.50i)13-s + (1.22 + 3.53i)14-s + (−0.0962 − 0.0962i)15-s + (1.36 + 3.76i)16-s + (−0.0852 − 0.318i)17-s + ⋯
L(s)  = 1  + (−0.301 + 0.953i)2-s + (−0.932 + 1.06i)3-s + (−0.818 − 0.574i)4-s + (−0.00162 + 0.0248i)5-s + (−0.733 − 1.20i)6-s + (0.802 − 0.596i)7-s + (0.794 − 0.607i)8-s + (−0.130 − 0.991i)9-s + (−0.0231 − 0.00901i)10-s + (0.592 − 1.74i)11-s + (1.37 − 0.334i)12-s + (0.622 − 0.416i)13-s + (0.326 + 0.945i)14-s + (−0.0248 − 0.0248i)15-s + (0.340 + 0.940i)16-s + (−0.0206 − 0.0771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.786 - 0.616i$
Analytic conductor: \(3.57729\)
Root analytic conductor: \(1.89137\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1/2),\ 0.786 - 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787503 + 0.271882i\)
\(L(\frac12)\) \(\approx\) \(0.787503 + 0.271882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.425 - 1.34i)T \)
7 \( 1 + (-2.12 + 1.57i)T \)
good3 \( 1 + (1.61 - 1.84i)T + (-0.391 - 2.97i)T^{2} \)
5 \( 1 + (0.00363 - 0.0554i)T + (-4.95 - 0.652i)T^{2} \)
11 \( 1 + (-1.96 + 5.78i)T + (-8.72 - 6.69i)T^{2} \)
13 \( 1 + (-2.24 + 1.50i)T + (4.97 - 12.0i)T^{2} \)
17 \( 1 + (0.0852 + 0.318i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.14 + 3.03i)T + (11.5 + 15.0i)T^{2} \)
23 \( 1 + (-0.893 - 6.78i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (-1.81 + 9.10i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (4.72 + 2.72i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.0177 + 0.271i)T + (-36.6 - 4.82i)T^{2} \)
41 \( 1 + (-9.05 - 3.74i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.291 + 1.46i)T + (-39.7 + 16.4i)T^{2} \)
47 \( 1 + (-0.342 + 1.27i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.74 - 2.96i)T + (42.0 + 32.2i)T^{2} \)
59 \( 1 + (-4.42 - 8.98i)T + (-35.9 + 46.8i)T^{2} \)
61 \( 1 + (9.59 - 3.25i)T + (48.3 - 37.1i)T^{2} \)
67 \( 1 + (6.31 + 5.53i)T + (8.74 + 66.4i)T^{2} \)
71 \( 1 + (2.36 + 5.71i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.58 - 4.67i)T + (-18.8 + 70.5i)T^{2} \)
79 \( 1 + (-1.11 + 4.17i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.38 + 4.93i)T + (31.7 - 76.6i)T^{2} \)
89 \( 1 + (-0.958 - 0.735i)T + (23.0 + 85.9i)T^{2} \)
97 \( 1 + 8.72T + 97T^{2} \)
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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

−10.95105728565762813540788758959, −10.48132089688222464703901664698, −9.255546462187233009786914472526, −8.554110960287474930888759402611, −7.53510553542079669773426665447, −6.18985444654146786330682313215, −5.68823356548461144215983469645, −4.57160285583982860276970501676, −3.80335908690754027522904357899, −0.75657423045446353955667419616, 1.39429466115159623646529258098, 2.19368484070746312152008001367, 4.21456684220929311406097325544, 5.10300927557249394356609142369, 6.46866227315031169948205208840, 7.26973713092530371787660199957, 8.452491653351249937681791870221, 9.134051661702951682714888749667, 10.54957529254590190783366541694, 11.01063723387262022374000421591

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