Properties

Degree 2
Conductor $ 17 \cdot 43 $
Sign $-0.911 - 0.410i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.41 + 1.41i)4-s + (−1.14 + 2.77i)9-s + (5.50 + 3.68i)11-s + (−4.77 − 4.77i)13-s − 4.00i·16-s + (−3.56 + 2.07i)17-s + (−0.868 + 1.29i)23-s + (−4.61 − 1.91i)25-s + (−8.40 + 5.61i)31-s + (−2.29 − 5.54i)36-s + (1.07 + 5.39i)41-s + (−2.50 + 6.05i)43-s + (−12.9 + 2.58i)44-s + (7.48 + 7.48i)47-s + (−6.46 + 2.67i)49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)4-s + (−0.382 + 0.923i)9-s + (1.66 + 1.11i)11-s + (−1.32 − 1.32i)13-s − 1.00i·16-s + (−0.864 + 0.502i)17-s + (−0.181 + 0.271i)23-s + (−0.923 − 0.382i)25-s + (−1.50 + 1.00i)31-s + (−0.382 − 0.923i)36-s + (0.167 + 0.842i)41-s + (−0.382 + 0.923i)43-s + (−1.95 + 0.389i)44-s + (1.09 + 1.09i)47-s + (−0.923 + 0.382i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.911 - 0.410i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(731\)    =    \(17 \cdot 43\)
\( \varepsilon \)  =  $-0.911 - 0.410i$
motivic weight  =  \(1\)
character  :  $\chi_{731} (515, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 731,\ (\ :1/2),\ -0.911 - 0.410i)$
$L(1)$  $\approx$  $0.132444 + 0.616418i$
$L(\frac12)$  $\approx$  $0.132444 + 0.616418i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{17,\;43\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{17,\;43\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad17 \( 1 + (3.56 - 2.07i)T \)
43 \( 1 + (2.50 - 6.05i)T \)
good2 \( 1 + (1.41 - 1.41i)T^{2} \)
3 \( 1 + (1.14 - 2.77i)T^{2} \)
5 \( 1 + (4.61 + 1.91i)T^{2} \)
7 \( 1 + (6.46 - 2.67i)T^{2} \)
11 \( 1 + (-5.50 - 3.68i)T + (4.20 + 10.1i)T^{2} \)
13 \( 1 + (4.77 + 4.77i)T + 13iT^{2} \)
19 \( 1 + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.868 - 1.29i)T + (-8.80 - 21.2i)T^{2} \)
29 \( 1 + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (8.40 - 5.61i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-1.07 - 5.39i)T + (-37.8 + 15.6i)T^{2} \)
47 \( 1 + (-7.48 - 7.48i)T + 47iT^{2} \)
53 \( 1 + (3.63 + 8.77i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (11.4 + 4.74i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + 11.7iT - 67T^{2} \)
71 \( 1 + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (67.4 + 27.9i)T^{2} \)
79 \( 1 + (-14.3 - 9.59i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (-6.05 + 2.50i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + 89iT^{2} \)
97 \( 1 + (-11.5 - 2.29i)T + (89.6 + 37.1i)T^{2} \)
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

−10.72453523824613955370462190335, −9.668141721124507719919141433693, −9.190336509228489074385349970208, −8.011837555802678078639393576732, −7.57359711081561629992050996415, −6.44191740459787805649206817794, −5.09469862273736637702291982912, −4.45645761126375441617577582771, −3.33068630543610846400243438958, −2.00124796174905465826421713184, 0.32101999828414267623015145419, 1.91139666797616460774983117229, 3.69965168637738892225575165052, 4.36974327993662498957692375522, 5.63810177256764021754439117865, 6.38378988141264223674221665301, 7.21091815044195470350881100406, 8.812875815952778765205749457660, 9.145537751088957674281534484263, 9.669963036802900276400966518302

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