↑

Properties

Label 2-448-112.13-c2-0-14
Degree $2$
Conductor $448$
Sign $0.850 - 0.525i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.99 + 2.99i)3-s + (0.657 + 0.657i)5-s + (−6.11 − 3.41i)7-s − 8.92i·9-s + (10.3 − 10.3i)11-s + (6.46 − 6.46i)13-s − 3.93·15-s + 12.4i·17-s + (9.56 − 9.56i)19-s + (28.5 − 8.06i)21-s + 19.8i·23-s − 24.1i·25-s + (−0.224 − 0.224i)27-s + (19.8 + 19.8i)29-s + 46.5i·31-s + ⋯
L(s)  = 1  + (−0.997 + 0.997i)3-s + (0.131 + 0.131i)5-s + (−0.872 − 0.487i)7-s − 0.991i·9-s + (0.939 − 0.939i)11-s + (0.497 − 0.497i)13-s − 0.262·15-s + 0.730i·17-s + (0.503 − 0.503i)19-s + (1.35 − 0.384i)21-s + 0.864i·23-s − 0.965i·25-s + (−0.00831 − 0.00831i)27-s + (0.683 + 0.683i)29-s + 1.50i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(448\)    =    \(2^{6} \cdot 7\)
Sign: $0.850 - 0.525i$
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: \((2,\ 448,\ (\ :1),\ 0.850 - 0.525i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.134015702\)
\(L(\frac12)\) \(\approx\) \(1.134015702\)
\(L(2)\) 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 \( 1 + (6.11 + 3.41i)T \)
good3 \( 1 + (2.99 - 2.99i)T - 9iT^{2} \)
5 \( 1 + (-0.657 - 0.657i)T + 25iT^{2} \)
11 \( 1 + (-10.3 + 10.3i)T - 121iT^{2} \)
13 \( 1 + (-6.46 + 6.46i)T - 169iT^{2} \)
17 \( 1 - 12.4iT - 289T^{2} \)
19 \( 1 + (-9.56 + 9.56i)T - 361iT^{2} \)
23 \( 1 - 19.8iT - 529T^{2} \)
29 \( 1 + (-19.8 - 19.8i)T + 841iT^{2} \)
31 \( 1 - 46.5iT - 961T^{2} \)
37 \( 1 + (14.3 - 14.3i)T - 1.36e3iT^{2} \)
41 \( 1 - 32.8T + 1.68e3T^{2} \)
43 \( 1 + (-35.2 + 35.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 36.8iT - 2.20e3T^{2} \)
53 \( 1 + (27.7 - 27.7i)T - 2.80e3iT^{2} \)
59 \( 1 + (-76.0 - 76.0i)T + 3.48e3iT^{2} \)
61 \( 1 + (-66.0 + 66.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-59.7 - 59.7i)T + 4.48e3iT^{2} \)
71 \( 1 + 39.5iT - 5.04e3T^{2} \)
73 \( 1 + 22.3T + 5.32e3T^{2} \)
79 \( 1 - 125.T + 6.24e3T^{2} \)
83 \( 1 + (-82.4 + 82.4i)T - 6.88e3iT^{2} \)
89 \( 1 + 41.7T + 7.92e3T^{2} \)
97 \( 1 + 115. iT - 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74557021129152039359137831903, −10.33245852606257189253682455277, −9.377140883373979072744574941404, −8.471634629468589500290974355417, −6.92241433407277600330907754040, −6.14157936235046715360586168448, −5.33471838332850161716246609303, −4.04756650678373979528897117046, −3.27870008845039239933064148850, −0.822383465402975711975543631867, 0.873559619154789801627176443953, 2.25519334988564900072672384277, 3.97243659307328338466782548864, 5.34946641916130176890708919462, 6.31692335431462414211996626676, 6.78447217099384354148991592819, 7.81239239520960760556652560331, 9.248076327120765525739716115250, 9.725714642308816268470369932073, 11.15923979869006052410514734082

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