Properties

Degree 2
Conductor $ 2^{3} \cdot 7^{2} $
Sign $-0.681 - 0.731i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.19 − 1.60i)2-s + (2.66 + 4.61i)3-s + (−1.13 + 3.83i)4-s + (−1.86 − 1.07i)5-s + (4.21 − 9.79i)6-s + (7.50 − 2.76i)8-s + (−9.71 + 16.8i)9-s + (0.506 + 4.28i)10-s + (−2.62 − 4.55i)11-s + (−20.7 + 4.97i)12-s + 21.4i·13-s − 11.5i·15-s + (−13.4 − 8.71i)16-s + (0.463 + 0.802i)17-s + (38.5 − 4.56i)18-s + (−2.96 + 5.13i)19-s + ⋯
L(s)  = 1  + (−0.598 − 0.801i)2-s + (0.888 + 1.53i)3-s + (−0.284 + 0.958i)4-s + (−0.373 − 0.215i)5-s + (0.701 − 1.63i)6-s + (0.938 − 0.345i)8-s + (−1.07 + 1.86i)9-s + (0.0506 + 0.428i)10-s + (−0.239 − 0.414i)11-s + (−1.72 + 0.414i)12-s + 1.64i·13-s − 0.766i·15-s + (−0.838 − 0.544i)16-s + (0.0272 + 0.0472i)17-s + (2.14 − 0.253i)18-s + (−0.156 + 0.270i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(392\)    =    \(2^{3} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.681 - 0.731i$
motivic weight  =  \(2\)
character  :  $\chi_{392} (275, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 392,\ (\ :1),\ -0.681 - 0.731i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.399030 + 0.917192i\)
\(L(\frac12)\)  \(\approx\)  \(0.399030 + 0.917192i\)
\(L(2)\)   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 \( 1 + (1.19 + 1.60i)T \)
7 \( 1 \)
good3 \( 1 + (-2.66 - 4.61i)T + (-4.5 + 7.79i)T^{2} \)
5 \( 1 + (1.86 + 1.07i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (2.62 + 4.55i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 21.4iT - 169T^{2} \)
17 \( 1 + (-0.463 - 0.802i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (2.96 - 5.13i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.52 - 4.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 9.42iT - 841T^{2} \)
31 \( 1 + (29.8 - 17.2i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (11.0 + 6.40i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + 43.1T + 1.68e3T^{2} \)
43 \( 1 + 41.7T + 1.84e3T^{2} \)
47 \( 1 + (39.8 + 22.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-64.5 + 37.2i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (-26.8 - 46.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-24.0 - 13.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-39.2 - 67.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 74.5iT - 5.04e3T^{2} \)
73 \( 1 + (-16.8 - 29.1i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (26.1 + 15.1i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 - 72.9T + 6.88e3T^{2} \)
89 \( 1 + (27.4 - 47.4i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 53.7T + 9.40e3T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−11.24584092580315821020577031179, −10.29524634936004919037501763221, −9.689101483346241192138642457076, −8.685081700062527555080811036178, −8.473807922039631216396657766277, −7.08001184906505551581048852074, −5.06645166064797448877748928051, −4.09126611643360700262711720423, −3.42950558381249588808895329904, −2.08427014658395446894495394240, 0.46561418950275031136336016445, 1.92872444509069939457528347960, 3.30495663046000457454151834557, 5.26355328368548934136211196511, 6.39349307720411956635842160812, 7.28937141202781996397574081046, 7.84160003475771248425005424981, 8.488440479587547297465443784226, 9.510106579694427851181553868054, 10.59815929744525236824037102850

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