Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.0815 + 0.996i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.86 − 0.722i)2-s + (−1.45 − 3.52i)3-s + (2.95 + 2.69i)4-s + (−3.21 + 7.75i)5-s + (0.174 + 7.62i)6-s + (1.87 − 1.87i)7-s + (−3.56 − 7.16i)8-s + (−3.92 + 3.92i)9-s + (11.5 − 12.1i)10-s + (3.78 − 9.14i)11-s + (5.18 − 14.3i)12-s + (6.24 + 15.0i)13-s + (−4.84 + 2.13i)14-s + 31.9·15-s + (1.46 + 15.9i)16-s − 19.0i·17-s + ⋯
L(s)  = 1  + (−0.932 − 0.361i)2-s + (−0.486 − 1.17i)3-s + (0.738 + 0.673i)4-s + (−0.642 + 1.55i)5-s + (0.0291 + 1.27i)6-s + (0.267 − 0.267i)7-s + (−0.445 − 0.895i)8-s + (−0.435 + 0.435i)9-s + (1.15 − 1.21i)10-s + (0.344 − 0.831i)11-s + (0.432 − 1.19i)12-s + (0.480 + 1.16i)13-s + (−0.345 + 0.152i)14-s + 2.13·15-s + (0.0916 + 0.995i)16-s − 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0815 + 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0815 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.0815 + 0.996i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (43, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.0815 + 0.996i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.494714 - 0.536853i\)
\(L(\frac12)\)  \(\approx\)  \(0.494714 - 0.536853i\)
\(L(2)\)   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 \( 1 + (1.86 + 0.722i)T \)
7 \( 1 + (-1.87 + 1.87i)T \)
good3 \( 1 + (1.45 + 3.52i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (3.21 - 7.75i)T + (-17.6 - 17.6i)T^{2} \)
11 \( 1 + (-3.78 + 9.14i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (-6.24 - 15.0i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 + (-24.1 + 10.0i)T + (255. - 255. i)T^{2} \)
23 \( 1 + (21.8 + 21.8i)T + 529iT^{2} \)
29 \( 1 + (-20.7 + 8.58i)T + (594. - 594. i)T^{2} \)
31 \( 1 - 13.8iT - 961T^{2} \)
37 \( 1 + (1.32 - 3.20i)T + (-968. - 968. i)T^{2} \)
41 \( 1 + (-31.4 + 31.4i)T - 1.68e3iT^{2} \)
43 \( 1 + (-22.9 + 55.4i)T + (-1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + 16.2T + 2.20e3T^{2} \)
53 \( 1 + (0.763 + 0.316i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (-70.5 - 29.2i)T + (2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-83.2 + 34.5i)T + (2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-4.73 - 11.4i)T + (-3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (4.10 - 4.10i)T - 5.04e3iT^{2} \)
73 \( 1 + (-55.6 + 55.6i)T - 5.32e3iT^{2} \)
79 \( 1 + 44.2T + 6.24e3T^{2} \)
83 \( 1 + (-29.0 + 12.0i)T + (4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (103. + 103. i)T + 7.92e3iT^{2} \)
97 \( 1 - 134.T + 9.40e3T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.53426281868240792083881342758, −11.16274432583440292608604839527, −10.01699575021614892271782556385, −8.667062630640964350649498026630, −7.48158730719076333001211512995, −6.98151540598878874640345975936, −6.24677517380448043526851135872, −3.72459923460097617560578981130, −2.40169742366650511273666124278, −0.68112869163914250812623540859, 1.24140911409848275624976279551, 3.93451311390664066314155912145, 5.10593225470753916887500032622, 5.81634815720502426865158801441, 7.75865179174117657658285258167, 8.344062983595436310848722002164, 9.497260149690880601337794993751, 10.01553409194013956801403771032, 11.19421313207115381352367134877, 11.96679309629843077154297990156

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