↑

Properties

Label 2-731-731.687-c1-0-58
Degree $2$
Conductor $731$
Sign $-0.911 + 0.410i$
Analytic cond. $5.83706$
Root an. cond. $2.41600$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

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

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.911 + 0.410i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.911 + 0.410i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.132444 - 0.616418i\)
\(L(\frac12)\) \(\approx\) \(0.132444 - 0.616418i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$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
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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