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

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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  =  $(2,\ 1792,\ (\ :1/2),\ -0.707 - 0.707i)$
$L(1)$  $\approx$  $1.486006467$
$L(\frac12)$  $\approx$  $1.486006467$
$L(\frac{3}{2})$   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 \( 1 + 10T + 97T^{2} \)
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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

−9.534787196938082010907880759058, −9.098578028097509415707346161483, −8.147052255068964491050557395691, −7.10244484516959338926717268054, −6.42260107548125418837606558146, −5.22685477394171817252048297576, −4.73233704940919964481830423819, −3.67724563045336390228385423503, −3.05986235449912193549632571266, −1.46859891296255733224064010658, 0.58402828276682568742413669358, 1.68724307891371138195539183749, 2.88170665607781651375695803383, 3.73128267301782271183106888680, 5.13872359531302107959415524867, 5.87383896144609479138352216976, 6.66590458914084282251461327277, 7.49402247491036111358051913974, 7.969755220518055100671528487094, 8.787882362901984389768593964098

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