Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $0.286 - 0.958i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.97 − 0.323i)2-s + (0.292 − 0.507i)3-s + (3.79 − 1.27i)4-s + (−7.82 + 4.51i)5-s + (0.414 − 1.09i)6-s + (7.07 − 3.74i)8-s + (4.32 + 7.49i)9-s + (−13.9 + 11.4i)10-s + (−6.24 + 10.8i)11-s + (0.463 − 2.29i)12-s + 9.03i·13-s + 5.29i·15-s + (12.7 − 9.66i)16-s + (6.17 − 10.6i)17-s + (10.9 + 13.3i)18-s + (14.4 + 25.0i)19-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)2-s + (0.0976 − 0.169i)3-s + (0.947 − 0.318i)4-s + (−1.56 + 0.903i)5-s + (0.0690 − 0.182i)6-s + (0.883 − 0.467i)8-s + (0.480 + 0.833i)9-s + (−1.39 + 1.14i)10-s + (−0.567 + 0.982i)11-s + (0.0386 − 0.191i)12-s + 0.694i·13-s + 0.352i·15-s + (0.796 − 0.604i)16-s + (0.363 − 0.628i)17-s + (0.609 + 0.744i)18-s + (0.759 + 1.31i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.286 - 0.958i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ 0.286 - 0.958i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.82672 + 1.36062i\)
\(L(\frac12)\)  \(\approx\)  \(1.82672 + 1.36062i\)
\(L(2)\)   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 \( 1 + (-1.97 + 0.323i)T \)
7 \( 1 \)
good3 \( 1 + (-0.292 + 0.507i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (7.82 - 4.51i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (6.24 - 10.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 9.03iT - 169T^{2} \)
17 \( 1 + (-6.17 + 10.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-14.4 - 25.0i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (21.3 - 12.3i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 22.4iT - 841T^{2} \)
31 \( 1 + (14.5 + 8.39i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-14.0 + 8.12i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 6.97T + 1.68e3T^{2} \)
43 \( 1 + 22.8T + 1.84e3T^{2} \)
47 \( 1 + (-5.36 + 3.09i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (6.94 + 4.00i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-15.2 + 26.3i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.1 + 7.61i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.3 + 68.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 17.5iT - 5.04e3T^{2} \)
73 \( 1 + (-23.3 + 40.4i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (70.1 - 40.5i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 40.3T + 6.88e3T^{2} \)
89 \( 1 + (-55.9 - 96.9i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 164.T + 9.40e3T^{2} \)
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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

−11.53160934096820881633165241173, −10.60511799086509601119787395938, −9.872546088226003653234592011014, −7.82095119242968588594283924412, −7.57065503127052998931880089069, −6.70465773631556104026272716113, −5.20136540836044121571370673706, −4.19982011388231300619091439003, −3.31939145905828591441183622329, −1.98366001459634990492369042085, 0.72870140849621422416125330254, 3.11331638665019045023412050970, 3.91594738059435677792295308182, 4.82319831115590960992665917584, 5.88528033147386695048673276755, 7.19257067835251871493145343379, 8.024530923846081648341447291574, 8.724409604956673301399901025472, 10.21638325247467376043492805516, 11.33445931813430819030988411375

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