Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.744 + 0.667i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.66 − 4.61i)3-s + (−1.86 + 1.07i)5-s + (−6.91 + 1.06i)7-s + (−9.71 − 16.8i)9-s + (2.62 − 4.55i)11-s − 21.4i·13-s + 11.5i·15-s + (−0.463 + 0.802i)17-s + (−2.96 − 5.13i)19-s + (−13.5 + 34.7i)21-s + (7.52 − 4.34i)23-s + (−10.1 + 17.6i)25-s − 55.6·27-s + 9.42i·29-s + (29.8 + 17.2i)31-s + ⋯
L(s)  = 1  + (0.888 − 1.53i)3-s + (−0.373 + 0.215i)5-s + (−0.988 + 0.152i)7-s + (−1.07 − 1.86i)9-s + (0.239 − 0.414i)11-s − 1.64i·13-s + 0.766i·15-s + (−0.0272 + 0.0472i)17-s + (−0.156 − 0.270i)19-s + (−0.644 + 1.65i)21-s + (0.327 − 0.188i)23-s + (−0.406 + 0.704i)25-s − 2.06·27-s + 0.324i·29-s + (0.963 + 0.556i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.744 + 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.744 + 0.667i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (207, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.744 + 0.667i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.538227 - 1.40724i\)
\(L(\frac12)\)  \(\approx\)  \(0.538227 - 1.40724i\)
\(L(2)\)   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 \( 1 + (6.91 - 1.06i)T \)
good3 \( 1 + (-2.66 + 4.61i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.86 - 1.07i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-2.62 + 4.55i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 + (0.463 - 0.802i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (2.96 + 5.13i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-7.52 + 4.34i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 9.42iT - 841T^{2} \)
31 \( 1 + (-29.8 - 17.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.0 + 6.40i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 43.1T + 1.68e3T^{2} \)
43 \( 1 - 41.7T + 1.84e3T^{2} \)
47 \( 1 + (-39.8 + 22.9i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (64.5 + 37.2i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-26.8 + 46.4i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-24.0 + 13.9i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (39.2 - 67.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (16.8 - 29.1i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (26.1 - 15.1i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 72.9T + 6.88e3T^{2} \)
89 \( 1 + (-27.4 - 47.4i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 53.7T + 9.40e3T^{2} \)
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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

−12.03633444112011077869943502012, −10.77974890369486788379827796743, −9.435469380946095881322072718186, −8.454425554311693461800966129793, −7.64408783603092455598625143076, −6.75586817795909675187139629017, −5.79080975913456633069507800729, −3.45668677393810565548505701874, −2.65715253184372298640245696898, −0.74487119924691309227906736146, 2.61157855964840144902095345486, 4.00410611935418369707450856146, 4.45358256496259743304262846348, 6.19258310179516389396867833527, 7.60632011368159216334086970476, 8.869464561110744238515548781704, 9.439943096237642270752412874304, 10.15609951414364042275635446933, 11.26014998569666464355879018390, 12.32151164209959648083614163200

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