Properties

Label 2-448-28.11-c2-0-25
Degree $2$
Conductor $448$
Sign $-0.895 + 0.444i$
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  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + 7i·7-s + (−4 − 6.92i)9-s + (−14.7 − 8.5i)11-s − 24·13-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (6.06 − 3.5i)19-s + (−3.5 + 6.06i)21-s + (6.06 − 3.5i)23-s + (12 − 20.7i)25-s − 17i·27-s − 24·29-s + (35.5 + 20.5i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.166i)3-s + (0.100 + 0.173i)5-s + i·7-s + (−0.444 − 0.769i)9-s + (−1.33 − 0.772i)11-s − 1.84·13-s + 0.0666i·15-s + (−0.0294 + 0.0509i)17-s + (0.319 − 0.184i)19-s + (−0.166 + 0.288i)21-s + (0.263 − 0.152i)23-s + (0.479 − 0.831i)25-s − 0.629i·27-s − 0.827·29-s + (1.14 + 0.661i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(12.2071\)
Root analytic conductor: \(3.49386\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{448} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 448,\ (\ :1),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1843515320\)
\(L(\frac12)\) \(\approx\) \(0.1843515320\)
\(L(2)\) 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 \( 1 - 7iT \)
good3 \( 1 + (-0.866 - 0.5i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (14.7 + 8.5i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 24T + 169T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-6.06 + 3.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 24T + 841T^{2} \)
31 \( 1 + (-35.5 - 20.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (24.5 + 42.4i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 48T + 1.68e3T^{2} \)
43 \( 1 - 24iT - 1.84e3T^{2} \)
47 \( 1 + (47.6 - 27.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (12.5 - 21.6i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-14.7 - 8.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-56.2 - 32.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 96iT - 5.04e3T^{2} \)
73 \( 1 + (47.5 - 82.2i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (35.5 - 20.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 72iT - 6.88e3T^{2} \)
89 \( 1 + (47.5 + 82.2i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 144T + 9.40e3T^{2} \)
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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

−10.36993619783146990051045197061, −9.586956248161989057344243953538, −8.707375864934206021625182393542, −7.919854473579603577832382696820, −6.73740712523444394382001175166, −5.62491079840176051253075442472, −4.85657467050835926375502413978, −3.09322934258198497851951944187, −2.48510538827698115864032007646, −0.06667605047837658086524589844, 1.97773768532242015162113459173, 3.09491007957084283877198376297, 4.78685603516214480234950170554, 5.21411894544555510434594442273, 7.01427346443869412904306104481, 7.54949489544941922417424132344, 8.326397731645163418798219129378, 9.786548939182096424755283664698, 10.15506427610640399083195906953, 11.18610155457664639166436839888

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