Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.916 + 0.400i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.46 + 1.36i)2-s + (−1.65 − 0.686i)3-s + (0.300 − 3.98i)4-s + (0.205 + 0.495i)5-s + (3.36 − 1.24i)6-s + (−0.193 + 6.99i)7-s + (4.98 + 6.25i)8-s + (−4.08 − 4.08i)9-s + (−0.974 − 0.447i)10-s + (−3.44 − 8.32i)11-s + (−3.23 + 6.40i)12-s + (−1.31 + 3.17i)13-s + (−9.23 − 10.5i)14-s − 0.962i·15-s + (−15.8 − 2.39i)16-s + 28.1·17-s + ⋯
L(s)  = 1  + (−0.733 + 0.680i)2-s + (−0.552 − 0.228i)3-s + (0.0751 − 0.997i)4-s + (0.0410 + 0.0990i)5-s + (0.560 − 0.207i)6-s + (−0.0276 + 0.999i)7-s + (0.623 + 0.782i)8-s + (−0.454 − 0.454i)9-s + (−0.0974 − 0.0447i)10-s + (−0.313 − 0.757i)11-s + (−0.269 + 0.533i)12-s + (−0.101 + 0.243i)13-s + (−0.659 − 0.751i)14-s − 0.0641i·15-s + (−0.988 − 0.149i)16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.916 + 0.400i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (13, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.916 + 0.400i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.744063 - 0.155477i\)
\(L(\frac12)\)  \(\approx\)  \(0.744063 - 0.155477i\)
\(L(2)\)   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 \( 1 + (1.46 - 1.36i)T \)
7 \( 1 + (0.193 - 6.99i)T \)
good3 \( 1 + (1.65 + 0.686i)T + (6.36 + 6.36i)T^{2} \)
5 \( 1 + (-0.205 - 0.495i)T + (-17.6 + 17.6i)T^{2} \)
11 \( 1 + (3.44 + 8.32i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (1.31 - 3.17i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 28.1T + 289T^{2} \)
19 \( 1 + (-12.1 + 29.2i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (11.0 + 11.0i)T + 529iT^{2} \)
29 \( 1 + (-6.55 + 15.8i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 31.5iT - 961T^{2} \)
37 \( 1 + (-39.9 + 16.5i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (-53.7 + 53.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-8.17 - 19.7i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 30.9T + 2.20e3T^{2} \)
53 \( 1 + (38.3 + 92.4i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (36.8 + 89.0i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-80.8 - 33.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (4.47 - 10.7i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (63.6 - 63.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (35.2 - 35.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 66.1iT - 6.24e3T^{2} \)
83 \( 1 + (-30.0 + 72.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (23.6 + 23.6i)T + 7.92e3iT^{2} \)
97 \( 1 - 115. iT - 9.40e3T^{2} \)
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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

−11.74375701373257195644017045100, −11.03910909740489098516080295203, −9.818492727708503927983902273579, −8.922345804880036098213347804664, −8.066609858103739322757049651409, −6.78438577861392188219246880434, −5.89203752908280390288542733117, −5.14624696956469660691536014757, −2.78364706974234337875268784303, −0.66662946092202651259955506463, 1.24153087912994893504295075585, 3.14543409691064473144598828312, 4.48132168059985281737463278620, 5.83882084607306153491043971405, 7.58268191419429410483215121795, 7.891192236138098809756705147131, 9.616253465089523233045281483081, 10.15740584250272240440979742383, 10.96144174850314727201472906307, 11.92152186069314253657962809772

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