Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.997 + 0.0684i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.20 − 1.27i)3-s + (3.56 + 2.05i)5-s + (6.98 − 0.407i)7-s + (−1.27 + 2.20i)9-s + (1.63 + 2.83i)11-s − 5.88i·13-s + 10.4·15-s + (−12.0 + 6.93i)17-s + (13.7 + 7.91i)19-s + (14.8 − 9.77i)21-s + (18.2 − 31.6i)23-s + (−4.01 − 6.95i)25-s + 29.3i·27-s − 28.4·29-s + (36.2 − 20.9i)31-s + ⋯
L(s)  = 1  + (0.733 − 0.423i)3-s + (0.713 + 0.411i)5-s + (0.998 − 0.0581i)7-s + (−0.141 + 0.244i)9-s + (0.148 + 0.257i)11-s − 0.452i·13-s + 0.697·15-s + (−0.707 + 0.408i)17-s + (0.721 + 0.416i)19-s + (0.707 − 0.465i)21-s + (0.793 − 1.37i)23-s + (−0.160 − 0.278i)25-s + 1.08i·27-s − 0.981·29-s + (1.16 − 0.674i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.997 + 0.0684i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.997 + 0.0684i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(2.32301 - 0.0796332i\)
\(L(\frac12)\)  \(\approx\)  \(2.32301 - 0.0796332i\)
\(L(2)\)   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 \( 1 \)
7 \( 1 + (-6.98 + 0.407i)T \)
good3 \( 1 + (-2.20 + 1.27i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.56 - 2.05i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 5.88iT - 169T^{2} \)
17 \( 1 + (12.0 - 6.93i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-13.7 - 7.91i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.2 + 31.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 28.4T + 841T^{2} \)
31 \( 1 + (-36.2 + 20.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (7.14 - 12.3i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 21.3iT - 1.68e3T^{2} \)
43 \( 1 + 55.3T + 1.84e3T^{2} \)
47 \( 1 + (29.3 + 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-42.4 - 73.4i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (58.5 - 33.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (25.6 + 14.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (27.4 + 47.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 83.8T + 5.04e3T^{2} \)
73 \( 1 + (108. - 62.8i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-35.1 + 60.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 27.1iT - 6.88e3T^{2} \)
89 \( 1 + (126. + 73.0i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 11.3iT - 9.40e3T^{2} \)
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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

−12.04848688351546669110475889308, −10.94364153221617510780080691741, −10.15292224034859420725190706502, −8.867119642562670114845952398779, −8.093719636664258229198607459813, −7.12395924975205286781283118457, −5.85542104780468219886415807899, −4.58307192475537716702574462280, −2.81175518254212283457069922435, −1.73311892041356998062349990124, 1.60236020686703517486543008118, 3.16335981417903371727982824187, 4.61399180746555828770986177316, 5.62249584297577547454384959850, 7.08142143071872739469728728748, 8.356373233394686852644711000351, 9.143645672033440621825589835236, 9.754141031180565314216334206050, 11.20820939359595998398689513289, 11.82014165044535365676777749769

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