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 + (0.292 − 0.507i)3-s + (−0.791 − 3.92i)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 + (−2.91 + 17.8i)10-s + (−6.24 + 10.8i)11-s + (−2.22 − 0.746i)12-s − 9.03i·13-s − 5.29i·15-s + (−14.7 + 6.20i)16-s + (6.17 − 10.6i)17-s + (−17.0 − 2.79i)18-s + (14.4 + 25.0i)19-s + ⋯
L(s)  = 1  + (−0.633 + 0.773i)2-s + (0.0976 − 0.169i)3-s + (−0.197 − 0.980i)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 + (−0.291 + 1.78i)10-s + (−0.567 + 0.982i)11-s + (−0.185 − 0.0622i)12-s − 0.694i·13-s − 0.352i·15-s + (−0.921 + 0.387i)16-s + (0.363 − 0.628i)17-s + (−0.949 − 0.155i)18-s + (0.759 + 1.31i)19-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $0.953 - 0.302i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (67, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ 0.953 - 0.302i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.70097 + 0.263171i\)
\(L(\frac12)\)  \(\approx\)  \(1.70097 + 0.263171i\)
\(L(2)\)   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 \( 1 + (-0.292 + 0.507i)T + (-4.5 - 7.79i)T^{2} \)
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

−10.52614851998127398247229187988, −10.01770388048993779309268648085, −9.409378677663631759177758784345, −8.276808971799471225561740901740, −7.55109991402735632011943565796, −6.39463672165096752683671426031, −5.25722258415722870070121375106, −4.92517241183119890257768516097, −2.26493459326438016052876621316, −1.20217601216040448057079297015, 1.26464481815968631257665005116, 2.65145611804196579478027461112, 3.48887820992028248865261312191, 5.20484075409393763447640545597, 6.48254150451861285965956050261, 7.22633052402357545275090215393, 8.753592154034289511052672502794, 9.393170047117965126767757595232, 10.11279466994730944782311956107, 10.85594005422532583684778237479

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