↑

Properties

Label 2-731-731.601-c1-0-16
Degree $2$
Conductor $731$
Sign $0.314 - 0.949i$
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 + (2.77 − 1.14i)9-s + (−0.955 + 4.80i)11-s + (−4.77 + 4.77i)13-s + 4.00i·16-s + (2.07 − 3.56i)17-s + (5.47 + 1.08i)23-s + (−1.91 − 4.61i)25-s + (0.749 + 3.76i)31-s + (5.54 + 2.29i)36-s + (−2.38 − 3.56i)41-s + (6.05 − 2.50i)43-s + (−8.14 + 5.44i)44-s + (−6.16 + 6.16i)47-s + (−2.67 + 6.46i)49-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)4-s + (0.923 − 0.382i)9-s + (−0.288 + 1.44i)11-s + (−1.32 + 1.32i)13-s + 1.00i·16-s + (0.502 − 0.864i)17-s + (1.14 + 0.227i)23-s + (−0.382 − 0.923i)25-s + (0.134 + 0.676i)31-s + (0.923 + 0.382i)36-s + (−0.371 − 0.556i)41-s + (0.923 − 0.382i)43-s + (−1.22 + 0.820i)44-s + (−0.898 + 0.898i)47-s + (−0.382 + 0.923i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41590 + 1.02221i\)
\(L(\frac12)\) \(\approx\) \(1.41590 + 1.02221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.07 + 3.56i)T \)
43 \( 1 + (-6.05 + 2.50i)T \)
good2 \( 1 + (-1.41 - 1.41i)T^{2} \)
3 \( 1 + (-2.77 + 1.14i)T^{2} \)
5 \( 1 + (1.91 + 4.61i)T^{2} \)
7 \( 1 + (2.67 - 6.46i)T^{2} \)
11 \( 1 + (0.955 - 4.80i)T + (-10.1 - 4.20i)T^{2} \)
13 \( 1 + (4.77 - 4.77i)T - 13iT^{2} \)
19 \( 1 + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-5.47 - 1.08i)T + (21.2 + 8.80i)T^{2} \)
29 \( 1 + (-11.0 - 26.7i)T^{2} \)
31 \( 1 + (-0.749 - 3.76i)T + (-28.6 + 11.8i)T^{2} \)
37 \( 1 + (-34.1 + 14.1i)T^{2} \)
41 \( 1 + (2.38 + 3.56i)T + (-15.6 + 37.8i)T^{2} \)
47 \( 1 + (6.16 - 6.16i)T - 47iT^{2} \)
53 \( 1 + (-10.1 - 4.22i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.46 - 8.36i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (-23.3 + 56.3i)T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + (-65.5 + 27.1i)T^{2} \)
73 \( 1 + (27.9 + 67.4i)T^{2} \)
79 \( 1 + (-2.96 + 14.9i)T + (-72.9 - 30.2i)T^{2} \)
83 \( 1 + (-2.50 + 6.05i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (16.2 + 10.8i)T + (37.1 + 89.6i)T^{2} \)
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

−10.45507773724410742443829885261, −9.713019219170829794839425870387, −9.003579524017565967058285029427, −7.41121384406685907807796203257, −7.36571372225256836925986243792, −6.50390540353131203251897460597, −4.88051286791954819606848631794, −4.22055936726821987043238808044, −2.80021182392715715590433274685, −1.80988955943073015625644373954, 0.926348790280399283107982104649, 2.39879415646494346275757477088, 3.48821797523684966364467641592, 5.11495284957919129656741418678, 5.60048506729065633123138691537, 6.73304007561082408087433347890, 7.58423323694122807828596365833, 8.330310116900740138981184434391, 9.708841875866702527877146475194, 10.23187490053831188709827379289

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