Properties

Degree 2
Conductor $ 2^{4} \cdot 7 $
Sign $0.382 - 0.923i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + i·7-s i·8-s i·9-s + (−1 − i)11-s − 14-s + 16-s + 18-s + (1 − i)22-s + i·25-s i·28-s + (−1 + i)29-s + i·32-s + i·36-s + (−1 − i)37-s + ⋯
L(s)  = 1  + i·2-s − 4-s + i·7-s i·8-s i·9-s + (−1 − i)11-s − 14-s + 16-s + 18-s + (1 − i)22-s + i·25-s i·28-s + (−1 + i)29-s + i·32-s + i·36-s + (−1 − i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(112\)    =    \(2^{4} \cdot 7\)
\( \varepsilon \)  =  $0.382 - 0.923i$
motivic weight  =  \(0\)
character  :  $\chi_{112} (69, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 112,\ (\ :0),\ 0.382 - 0.923i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.5314225250\)
\(L(\frac12)\)  \(\approx\)  \(0.5314225250\)
\(L(1)\)   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 \( 1 - iT \)
7 \( 1 - iT \)
good3 \( 1 + iT^{2} \)
5 \( 1 - iT^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (1 - i)T - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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

−14.30069270320800848660558005175, −13.13005281311943114418937395804, −12.31230408122088277357194352773, −10.90122155092304599712070645171, −9.372527297341524736564459728269, −8.712784745011879189339577700650, −7.48250950853313495465445459749, −6.09862088320331059470531847771, −5.28799620715987171801146708145, −3.39688143770569238837513800590, 2.27803974715281569194064287612, 4.13114150743211819775629176162, 5.22631067709461204170447629166, 7.34148884483076194341677592755, 8.319056652516069500791959709910, 9.968337038803622567918569441583, 10.42149304749246711383091061266, 11.49301810311495345341512613038, 12.75795018895155456044755509988, 13.45286471755867347856734524811

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