Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.0284 - 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0284 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0284 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.0284 - 0.999i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (79, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.0284 - 0.999i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.59848 + 1.64466i\)
\(L(\frac12)\)  \(\approx\)  \(1.59848 + 1.64466i\)
\(L(2)\)   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

−12.20115487321573048964937160338, −10.78701379472959123998386417840, −10.40465382340424726584808896483, −9.398684209123105421164029574769, −8.472411464142201851907023710639, −7.67153107644832192454578473346, −5.69373357455032486373619856540, −4.76782109550315070214078409088, −3.61655452841785874109024417000, −2.32548370289706160110479165417, 1.36605083453434921383499543757, 2.27929703696035106858965262291, 4.06838734497075548737398921014, 5.81112133895433562161902907628, 6.93867776687590700309196144971, 7.72759876730605609046999434724, 8.724298112190728403613461529375, 9.380599852958993922976194152494, 11.20330668796126021811617376373, 11.85484371895990751730700320714

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