Properties

Degree 2
Conductor $ 2^{5} \cdot 7 $
Sign $0.924 - 0.380i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 - 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(224\)    =    \(2^{5} \cdot 7\)
\( \varepsilon \)  =  $0.924 - 0.380i$
motivic weight  =  \(2\)
character  :  $\chi_{224} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 224,\ (\ :1),\ 0.924 - 0.380i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.72355 + 0.340508i\)
\(L(\frac12)\)  \(\approx\)  \(1.72355 + 0.340508i\)
\(L(2)\)   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

−12.11056289385182420692060151772, −10.77280031200866007560520274241, −10.41683479564898727616025270013, −9.275036934421825199442911499368, −7.973784744384075454339995714240, −6.96054463207658255992044078092, −6.14255073525375109112185483551, −4.51728160425399173815902188668, −3.32329173685600821829238845250, −1.52626448428376685626714303281, 1.20478158324026177384312385866, 3.12531932009444758080935433931, 4.47734145430205568470529064584, 5.94346278856885881735234639056, 6.56039741751943090548011683765, 8.279477861876901417908267243778, 9.169207872549241901872289855087, 9.648306912592559048009181151655, 11.22359196384740848271031478532, 11.91400896529693015742346828263

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