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Properties

Label 2-731-731.16-c1-0-61
Degree $2$
Conductor $731$
Sign $-0.321 - 0.946i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.501 − 2.19i)2-s + (1.02 + 0.234i)3-s + (−2.77 + 1.33i)4-s + (−1.62 + 1.29i)5-s − 2.37i·6-s − 4.11i·7-s + (1.50 + 1.89i)8-s + (−1.70 − 0.819i)9-s + (3.66 + 2.91i)10-s + (0.236 − 0.491i)11-s + (−3.15 + 0.721i)12-s + (2.43 + 3.05i)13-s + (−9.04 + 2.06i)14-s + (−1.97 + 0.950i)15-s + (−0.433 + 0.542i)16-s + (−2.62 − 3.17i)17-s + ⋯
L(s)  = 1  + (−0.354 − 1.55i)2-s + (0.593 + 0.135i)3-s + (−1.38 + 0.667i)4-s + (−0.726 + 0.579i)5-s − 0.969i·6-s − 1.55i·7-s + (0.533 + 0.669i)8-s + (−0.567 − 0.273i)9-s + (1.15 + 0.923i)10-s + (0.0713 − 0.148i)11-s + (−0.912 + 0.208i)12-s + (0.675 + 0.847i)13-s + (−2.41 + 0.551i)14-s + (−0.509 + 0.245i)15-s + (−0.108 + 0.135i)16-s + (−0.636 − 0.771i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.321 - 0.946i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255537 + 0.356670i\)
\(L(\frac12)\) \(\approx\) \(0.255537 + 0.356670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.62 + 3.17i)T \)
43 \( 1 + (5.74 - 3.15i)T \)
good2 \( 1 + (0.501 + 2.19i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-1.02 - 0.234i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (1.62 - 1.29i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 + 4.11iT - 7T^{2} \)
11 \( 1 + (-0.236 + 0.491i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.43 - 3.05i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (1.60 - 0.772i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (-0.956 + 1.98i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (3.76 - 0.859i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (6.82 - 1.55i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 - 0.516iT - 37T^{2} \)
41 \( 1 + (4.31 - 0.985i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (7.94 - 3.82i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-8.64 + 10.8i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (8.03 - 10.0i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-7.03 - 1.60i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (-8.76 + 4.22i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (5.92 + 12.3i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-6.97 + 5.56i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 - 2.79iT - 79T^{2} \)
83 \( 1 + (-2.21 + 9.68i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-3.87 + 16.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-5.07 + 10.5i)T + (-60.4 - 75.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.949755950526606007367639879652, −9.126202819870742982107891434029, −8.417820701584070720872307781840, −7.34380374317320578511864200825, −6.52658967903377373564283937905, −4.53845442865765755196434607946, −3.64284459313026300832832277484, −3.27464998041663678147982236171, −1.81278127594276720332629719657, −0.22728425601800881189118977798, 2.26797232287015942208102639249, 3.71938045094505227675665324773, 5.19683555778703133960980789341, 5.66838113340999135289308893493, 6.66754916344133146235435798913, 7.84888144167178074961600880321, 8.397718741953834251446105245719, 8.730262269195099021878427516729, 9.493716071705853875845401641239, 11.00472863916339572653761911567

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