Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.948 - 0.317i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)2-s + (−2.79 − 1.15i)3-s + (−0.0129 − 3.99i)4-s + (−1.86 − 4.49i)5-s + (5.58 − 2.32i)6-s + (0.944 − 6.93i)7-s + (5.68 + 5.62i)8-s + (0.104 + 0.104i)9-s + (8.99 + 3.70i)10-s + (2.63 + 6.36i)11-s + (−4.59 + 11.1i)12-s + (−3.81 + 9.20i)13-s + (8.49 + 11.1i)14-s + 14.7i·15-s + (−15.9 + 0.103i)16-s − 18.4·17-s + ⋯
L(s)  = 1  + (−0.705 + 0.708i)2-s + (−0.931 − 0.385i)3-s + (−0.00324 − 0.999i)4-s + (−0.372 − 0.898i)5-s + (0.930 − 0.387i)6-s + (0.134 − 0.990i)7-s + (0.710 + 0.703i)8-s + (0.0115 + 0.0115i)9-s + (0.899 + 0.370i)10-s + (0.239 + 0.578i)11-s + (−0.382 + 0.932i)12-s + (−0.293 + 0.708i)13-s + (0.606 + 0.795i)14-s + 0.980i·15-s + (−0.999 + 0.00649i)16-s − 1.08·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.948 - 0.317i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.948 - 0.317i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.00895650 + 0.0548803i\)
\(L(\frac12)\)  \(\approx\)  \(0.00895650 + 0.0548803i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 1.41i)T \)
7 \( 1 + (-0.944 + 6.93i)T \)
good3 \( 1 + (2.79 + 1.15i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (1.86 + 4.49i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (-2.63 - 6.36i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (3.81 - 9.20i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 18.4T + 289T^{2} \)
19 \( 1 + (-6.94 + 16.7i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (1.72 + 1.72i)T + 529iT^{2} \)
29 \( 1 + (17.1 - 41.4i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 2.69iT - 961T^{2} \)
37 \( 1 + (63.1 - 26.1i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-4.62 + 4.62i)T - 1.68e3iT^{2} \)
43 \( 1 + (-1.15 - 2.78i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 9.80T + 2.20e3T^{2} \)
53 \( 1 + (-10.8 - 26.2i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-30.6 - 73.9i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-25.7 - 10.6i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-29.2 + 70.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-33.8 + 33.8i)T - 5.04e3iT^{2} \)
73 \( 1 + (70.7 - 70.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 89.2iT - 6.24e3T^{2} \)
83 \( 1 + (54.0 - 130. i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-72.3 - 72.3i)T + 7.92e3iT^{2} \)
97 \( 1 + 108. iT - 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.37711234564205870683970248964, −10.58919631795234855009374731723, −9.312281360329363273227764748386, −8.549424363203081126508110010099, −7.09737776843279837935341993727, −6.78952782844003575254547137361, −5.23835711676191279052312116181, −4.44106454830355300145474049120, −1.36602519029325736261555170963, −0.04538633463181669397646998618, 2.39174041655241606371667287577, 3.69950140838623515525796217981, 5.30124268389589127372333039014, 6.45941013199546994347073154045, 7.78144796794242193576722753302, 8.746078923191707369343379225439, 9.938441471525064618878365446403, 10.79156564271557050734987430119, 11.45075709637212938424517273544, 11.99541798102874118286576155679

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