Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $-0.754 + 0.656i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.126 + 0.219i)3-s + (−1.78 − 3.09i)5-s + (−2.89 − 6.37i)7-s + (4.46 + 7.73i)9-s + (−6.82 − 3.94i)11-s − 18.1·13-s + 0.904·15-s + (−8.26 − 4.76i)17-s + (−12.4 − 21.4i)19-s + (1.76 + 0.172i)21-s + (−2.14 − 3.72i)23-s + (6.12 − 10.6i)25-s − 4.54·27-s − 28.3i·29-s + (28.2 + 16.3i)31-s + ⋯
L(s)  = 1  + (−0.0422 + 0.0731i)3-s + (−0.357 − 0.618i)5-s + (−0.413 − 0.910i)7-s + (0.496 + 0.859i)9-s + (−0.620 − 0.358i)11-s − 1.39·13-s + 0.0603·15-s + (−0.485 − 0.280i)17-s + (−0.653 − 1.13i)19-s + (0.0840 + 0.00819i)21-s + (−0.0934 − 0.161i)23-s + (0.244 − 0.424i)25-s − 0.168·27-s − 0.978i·29-s + (0.910 + 0.525i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $-0.754 + 0.656i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ -0.754 + 0.656i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.238205 - 0.636667i\)
\(L(\frac12)\)  \(\approx\)  \(0.238205 - 0.636667i\)
\(L(2)\)   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 \( 1 + (2.89 + 6.37i)T \)
good3 \( 1 + (0.126 - 0.219i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (1.78 + 3.09i)T + (-12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.82 + 3.94i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 18.1T + 169T^{2} \)
17 \( 1 + (8.26 + 4.76i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (12.4 + 21.4i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (2.14 + 3.72i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + 28.3iT - 841T^{2} \)
31 \( 1 + (-28.2 - 16.3i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (25.9 - 14.9i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 45.2iT - 1.68e3T^{2} \)
43 \( 1 - 24.9iT - 1.84e3T^{2} \)
47 \( 1 + (44.0 - 25.4i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-54.3 - 31.4i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-37.0 + 64.0i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-25.2 - 43.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-108. - 62.7i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.33T + 5.04e3T^{2} \)
73 \( 1 + (23.6 + 13.6i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (51.5 + 89.2i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 51.5T + 6.88e3T^{2} \)
89 \( 1 + (-133. + 76.9i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 47.0iT - 9.40e3T^{2} \)
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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

−11.65016204423494787185294051161, −10.55007975845684807061626595171, −9.909087702793595058271406829785, −8.600706330003159052258140711299, −7.61526273865759447374343678277, −6.74423616383035093686089258256, −4.99903116610057123566450411734, −4.35790146229950679297516361587, −2.54316407126107616851632084985, −0.35142182324405637853888713464, 2.29614013162962967126636398135, 3.62095922543947045819814689739, 5.11553376144193165932815749542, 6.42373852013697328709714961728, 7.25010861708573691978557241178, 8.444395663065281104364274505307, 9.642454335602233933676885658175, 10.30910121133134136902497500892, 11.62207755706264433277123132342, 12.40843316288848548298419112965

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