↑

Properties

Label 2-731-17.9-c1-0-31
Degree $2$
Conductor $731$
Sign $-0.0822 - 0.996i$
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  + (−0.292 + 0.292i)2-s + (2.84 + 1.17i)3-s + 1.82i·4-s + (0.0154 − 0.0374i)5-s + (−1.17 + 0.488i)6-s + (−0.341 − 0.825i)7-s + (−1.12 − 1.12i)8-s + (4.58 + 4.58i)9-s + (0.00641 + 0.0154i)10-s + (2.69 − 1.11i)11-s + (−2.15 + 5.20i)12-s + 4.69i·13-s + (0.341 + 0.141i)14-s + (0.0881 − 0.0881i)15-s − 3.00·16-s + (−0.147 − 4.12i)17-s + ⋯
L(s)  = 1  + (−0.207 + 0.207i)2-s + (1.64 + 0.680i)3-s + 0.914i·4-s + (0.00693 − 0.0167i)5-s + (−0.480 + 0.199i)6-s + (−0.129 − 0.311i)7-s + (−0.396 − 0.396i)8-s + (1.52 + 1.52i)9-s + (0.00202 + 0.00490i)10-s + (0.813 − 0.337i)11-s + (−0.621 + 1.50i)12-s + 1.30i·13-s + (0.0913 + 0.0378i)14-s + (0.0227 − 0.0227i)15-s − 0.750·16-s + (−0.0357 − 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0822 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.0822 - 0.996i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.0822 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59673 + 1.73386i\)
\(L(\frac12)\) \(\approx\) \(1.59673 + 1.73386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.147 + 4.12i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
good2 \( 1 + (0.292 - 0.292i)T - 2iT^{2} \)
3 \( 1 + (-2.84 - 1.17i)T + (2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.0154 + 0.0374i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (0.341 + 0.825i)T + (-4.94 + 4.94i)T^{2} \)
11 \( 1 + (-2.69 + 1.11i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 - 4.69iT - 13T^{2} \)
19 \( 1 + (-1.40 + 1.40i)T - 19iT^{2} \)
23 \( 1 + (7.48 - 3.09i)T + (16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.652 + 1.57i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + (-6.85 - 2.83i)T + (21.9 + 21.9i)T^{2} \)
37 \( 1 + (8.16 + 3.38i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (0.849 + 2.05i)T + (-28.9 + 28.9i)T^{2} \)
47 \( 1 + 13.0iT - 47T^{2} \)
53 \( 1 + (9.03 - 9.03i)T - 53iT^{2} \)
59 \( 1 + (1.17 + 1.17i)T + 59iT^{2} \)
61 \( 1 + (5.00 + 12.0i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + 4.59T + 67T^{2} \)
71 \( 1 + (-12.1 - 5.03i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (3.38 - 8.18i)T + (-51.6 - 51.6i)T^{2} \)
79 \( 1 + (-7.47 + 3.09i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (-8.05 + 8.05i)T - 83iT^{2} \)
89 \( 1 + 9.27iT - 89T^{2} \)
97 \( 1 + (-1.77 + 4.27i)T + (-68.5 - 68.5i)T^{2} \)
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

−10.24569592298241865793297935134, −9.294995220221859520821927022878, −9.049696734466100719537356792031, −8.200191184773425318095224740286, −7.34430448555359941904697058227, −6.66214783742284934118963876724, −4.75820680877158746482088759079, −3.84805545602058024706225619220, −3.28069104540291649002155699466, −2.06182942417865690306677522901, 1.22197799005775682044354921978, 2.24808669869308820681242377828, 3.22804450030838594954118313512, 4.46190445275558983718504630017, 6.00850024131547862743719016095, 6.63376004191565754922333768927, 7.951456424055028217698659751531, 8.382687775174237765599119130908, 9.287376780687851265961902869764, 9.986586968517400047398577786708

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