Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.0104 - 0.999i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.19 + 1.84i)3-s + (−2.63 + 1.52i)5-s + (−0.812 + 6.95i)7-s + (2.28 + 3.96i)9-s + (1.17 − 2.03i)11-s + 25.3i·13-s − 11.2·15-s + (3.08 + 1.78i)17-s + (14.1 − 8.18i)19-s + (−15.4 + 20.6i)21-s + (−8.83 − 15.3i)23-s + (−7.85 + 13.6i)25-s − 16.2i·27-s + 36.1·29-s + (−6.25 − 3.61i)31-s + ⋯
L(s)  = 1  + (1.06 + 0.614i)3-s + (−0.527 + 0.304i)5-s + (−0.116 + 0.993i)7-s + (0.254 + 0.440i)9-s + (0.106 − 0.185i)11-s + 1.94i·13-s − 0.748·15-s + (0.181 + 0.104i)17-s + (0.746 − 0.430i)19-s + (−0.733 + 0.985i)21-s + (−0.384 − 0.665i)23-s + (−0.314 + 0.544i)25-s − 0.603i·27-s + 1.24·29-s + (−0.201 − 0.116i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0104 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0104 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.0104 - 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (129, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.0104 - 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.31901 + 1.33281i\)
\(L(\frac12)\)  \(\approx\)  \(1.31901 + 1.33281i\)
\(L(2)\)   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 \( 1 + (0.812 - 6.95i)T \)
good3 \( 1 + (-3.19 - 1.84i)T + (4.5 + 7.79i)T^{2} \)
5 \( 1 + (2.63 - 1.52i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-1.17 + 2.03i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 25.3iT - 169T^{2} \)
17 \( 1 + (-3.08 - 1.78i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-14.1 + 8.18i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (8.83 + 15.3i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 36.1T + 841T^{2} \)
31 \( 1 + (6.25 + 3.61i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (18.4 + 31.8i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 53.7iT - 1.68e3T^{2} \)
43 \( 1 - 51.2T + 1.84e3T^{2} \)
47 \( 1 + (-27.1 + 15.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-35.1 + 60.8i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-81.4 - 47.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.89 - 1.09i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-12.4 + 21.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 50.8T + 5.04e3T^{2} \)
73 \( 1 + (68.9 + 39.7i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (57.5 + 99.6i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 154. iT - 6.88e3T^{2} \)
89 \( 1 + (-98.7 + 57.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 53.9iT - 9.40e3T^{2} \)
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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

−11.98500986736126657025853156759, −11.52724005384875127077115800465, −10.08766972282007489026795148334, −9.090187385721957798622205383541, −8.694480391946553371804053139218, −7.39544112476594388825976999883, −6.17929380447177145438480816800, −4.56529802896818063564463764013, −3.49768904699806584353222632190, −2.31429007654766115830924356662, 0.971338917017686934594998859805, 2.86923057644656756638241113287, 3.92499782225769252404026180103, 5.49815689981094920494470470305, 7.16242884192736948480481287221, 7.83577994919443310509224763977, 8.468523320299974624038623323917, 9.855701221058309155403508564585, 10.66057267699072258946783664834, 12.07197262148970441776830902128

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