Properties

Degree 2
Conductor $ 2^{8} \cdot 7 $
Sign $0.707 + 0.707i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·3-s + 7-s − 9-s + 4i·13-s + 6·17-s + 2i·19-s − 2i·21-s + 5·25-s − 4i·27-s + 6i·29-s + 4·31-s + 2i·37-s + 8·39-s − 6·41-s − 8i·43-s + ⋯
L(s)  = 1  − 1.15i·3-s + 0.377·7-s − 0.333·9-s + 1.10i·13-s + 1.45·17-s + 0.458i·19-s − 0.436i·21-s + 25-s − 0.769i·27-s + 1.11i·29-s + 0.718·31-s + 0.328i·37-s + 1.28·39-s − 0.937·41-s − 1.21i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1792\)    =    \(2^{8} \cdot 7\)
\( \varepsilon \)  =  $0.707 + 0.707i$
motivic weight  =  \(1\)
character  :  $\chi_{1792} (897, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 1792,\ (\ :1/2),\ 0.707 + 0.707i)$
$L(1)$  $\approx$  $1.988236859$
$L(\frac12)$  $\approx$  $1.988236859$
$L(\frac{3}{2})$   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 \( 1 \)
7 \( 1 - T \)
good3 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 12T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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

−8.969894990703285873460521721506, −8.319348796562183905995251752788, −7.42972218565900774766647327548, −6.98830406054245652288390218989, −6.12207400655819751030723132536, −5.22322005488711877978514198455, −4.20828599827424517765801743325, −3.05346149573610183813753309475, −1.85911327474741436580407073794, −1.07240091572505906501474742470, 1.00373239285767889559039585838, 2.70743521323646971641217695588, 3.53830435246007457620500860630, 4.47867284023329427029777746450, 5.20561269830225679526579033922, 5.89274378722533440007963278070, 7.10860822769963937535741884241, 7.942853529295811248914099066303, 8.629298718029248324586048066036, 9.582244117207813306484081522867

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