Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $-0.575 - 0.817i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.51 + 1.30i)2-s + (0.824 − 1.42i)3-s + (0.597 + 3.95i)4-s + (−3.95 + 2.28i)5-s + (3.11 − 1.08i)6-s + (−4.25 + 6.77i)8-s + (3.14 + 5.43i)9-s + (−8.96 − 1.69i)10-s + (6.18 − 10.7i)11-s + (6.13 + 2.40i)12-s + 18.3i·13-s + 7.52i·15-s + (−15.2 + 4.72i)16-s + (−6.51 + 11.2i)17-s + (−2.33 + 12.3i)18-s + (−1.51 − 2.61i)19-s + ⋯
L(s)  = 1  + (0.758 + 0.652i)2-s + (0.274 − 0.475i)3-s + (0.149 + 0.988i)4-s + (−0.790 + 0.456i)5-s + (0.518 − 0.181i)6-s + (−0.531 + 0.846i)8-s + (0.348 + 0.604i)9-s + (−0.896 − 0.169i)10-s + (0.562 − 0.974i)11-s + (0.511 + 0.200i)12-s + 1.41i·13-s + 0.501i·15-s + (−0.955 + 0.295i)16-s + (−0.383 + 0.663i)17-s + (−0.129 + 0.685i)18-s + (−0.0796 − 0.137i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.575 - 0.817i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ -0.575 - 0.817i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.996610 + 1.92044i\)
\(L(\frac12)\)  \(\approx\)  \(0.996610 + 1.92044i\)
\(L(2)\)   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 \( 1 + (-1.51 - 1.30i)T \)
7 \( 1 \)
good3 \( 1 + (-0.824 + 1.42i)T + (-4.5 - 7.79i)T^{2} \)
5 \( 1 + (3.95 - 2.28i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (-6.18 + 10.7i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 18.3iT - 169T^{2} \)
17 \( 1 + (6.51 - 11.2i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (1.51 + 2.61i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (26.2 - 15.1i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 22.7iT - 841T^{2} \)
31 \( 1 + (19.5 + 11.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-11.9 + 6.88i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 60.5T + 1.68e3T^{2} \)
43 \( 1 - 39.0T + 1.84e3T^{2} \)
47 \( 1 + (-17.6 + 10.1i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-4.12 - 2.38i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-5.86 + 10.1i)T + (-1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-94.3 + 54.4i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-39.5 + 68.5i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 12.9iT - 5.04e3T^{2} \)
73 \( 1 + (49.2 - 85.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-113. + 65.6i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 28.3T + 6.88e3T^{2} \)
89 \( 1 + (78.7 + 136. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 39.6T + 9.40e3T^{2} \)
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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

−11.45998768413458706930125103488, −10.98040335591103971806555564729, −9.246075147938435967419814999625, −8.309268193059275307703158026374, −7.50514904595532418148311368789, −6.77479771861542638988843116820, −5.81210383292442407364936822718, −4.31672862180237015938672495221, −3.61619363295148891416728908984, −2.08572570824559889981556375703, 0.72515083703637123421572838954, 2.56959022304154785556906304290, 3.96536272307679868898887602130, 4.36569886678398260157052750218, 5.65685604412218879580682217941, 6.87794864406011498388876506930, 8.048515687332462877590918115158, 9.265432329822885701293098551515, 9.943197938218288323625601583600, 10.81253167382421237773192713653

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