Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $-0.953 - 0.302i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.26 + 1.54i)2-s + (1.70 + 2.95i)3-s + (−0.791 + 3.92i)4-s + (1.34 + 0.774i)5-s + (−2.41 + 6.38i)6-s + (−7.07 + 3.74i)8-s + (−1.32 + 2.30i)9-s + (0.500 + 3.05i)10-s + (2.24 + 3.88i)11-s + (−12.9 + 4.35i)12-s + 1.54i·13-s + 5.29i·15-s + (−14.7 − 6.20i)16-s + (11.8 + 20.4i)17-s + (−5.24 + 0.858i)18-s + (−12.4 + 21.5i)19-s + ⋯
L(s)  = 1  + (0.633 + 0.773i)2-s + (0.569 + 0.985i)3-s + (−0.197 + 0.980i)4-s + (0.268 + 0.154i)5-s + (−0.402 + 1.06i)6-s + (−0.883 + 0.467i)8-s + (−0.147 + 0.255i)9-s + (0.0500 + 0.305i)10-s + (0.203 + 0.353i)11-s + (−1.07 + 0.362i)12-s + 0.119i·13-s + 0.352i·15-s + (−0.921 − 0.387i)16-s + (0.695 + 1.20i)17-s + (−0.291 + 0.0476i)18-s + (−0.654 + 1.13i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.953 - 0.302i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (275, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ -0.953 - 0.302i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.418802 + 2.70687i\)
\(L(\frac12)\)  \(\approx\)  \(0.418802 + 2.70687i\)
\(L(2)\)   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 \( 1 + (-1.26 - 1.54i)T \)
7 \( 1 \)
good3 \( 1 + (-1.70 - 2.95i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (-1.34 - 0.774i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-2.24 - 3.88i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 1.54iT - 169T^{2} \)
17 \( 1 + (-11.8 - 20.4i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (12.4 - 21.5i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (30.5 + 17.6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 22.4iT - 841T^{2} \)
31 \( 1 + (-40.4 + 23.3i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-50.7 - 29.2i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 26.9T + 1.68e3T^{2} \)
43 \( 1 + 17.1T + 1.84e3T^{2} \)
47 \( 1 + (31.2 + 18.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-84.7 + 48.9i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-30.7 - 53.3i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (32.6 + 18.8i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.6 - 28.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 + (-34.6 - 60.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (33.5 + 19.3i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 3.61T + 6.88e3T^{2} \)
89 \( 1 + (-22.0 + 38.1i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + 96.1T + 9.40e3T^{2} \)
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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

−11.72554540882562729929322593416, −10.11277678753393865379865108073, −9.890060474805879596716282723242, −8.418002819729893878134047693996, −8.059306049284942663738411726693, −6.48663082250791939704826592731, −5.84479181294071758968926908521, −4.27357021044391494376290438689, −3.96128297498935552097060277628, −2.46774194023061024667783146000, 0.959894915781583229640164619873, 2.20546257870998399001972504342, 3.20444699263655240049266260714, 4.65081628809979034074334597713, 5.76551934850402190721483705653, 6.82762499237317566502413073896, 7.84743681547816135769524718179, 8.997926550343008254930542801675, 9.773968601194952906893176703269, 10.89153142589370027853954091387

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