Properties

Label 2-731-43.6-c1-0-37
Degree $2$
Conductor $731$
Sign $0.942 + 0.333i$
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  + 0.715·2-s + (−1.27 + 2.21i)3-s − 1.48·4-s + (0.380 − 0.658i)5-s + (−0.915 + 1.58i)6-s + (−1.25 − 2.16i)7-s − 2.49·8-s + (−1.77 − 3.06i)9-s + (0.272 − 0.471i)10-s + 3.64·11-s + (1.90 − 3.29i)12-s + (−0.946 − 1.63i)13-s + (−0.895 − 1.55i)14-s + (0.971 + 1.68i)15-s + 1.18·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  + 0.506·2-s + (−0.738 + 1.27i)3-s − 0.743·4-s + (0.169 − 0.294i)5-s + (−0.373 + 0.647i)6-s + (−0.473 − 0.819i)7-s − 0.882·8-s + (−0.590 − 1.02i)9-s + (0.0860 − 0.149i)10-s + 1.09·11-s + (0.549 − 0.951i)12-s + (−0.262 − 0.454i)13-s + (−0.239 − 0.414i)14-s + (0.250 + 0.434i)15-s + 0.296·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.942 + 0.333i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03602 - 0.177888i\)
\(L(\frac12)\) \(\approx\) \(1.03602 - 0.177888i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-5.24 - 3.94i)T \)
good2 \( 1 - 0.715T + 2T^{2} \)
3 \( 1 + (1.27 - 2.21i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.380 + 0.658i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.25 + 2.16i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.64T + 11T^{2} \)
13 \( 1 + (0.946 + 1.63i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-3.30 + 5.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 - 2.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.32 + 2.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.99 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.162 + 0.280i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.56T + 41T^{2} \)
47 \( 1 + 1.45T + 47T^{2} \)
53 \( 1 + (-4.87 + 8.45i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.5T + 59T^{2} \)
61 \( 1 + (4.32 + 7.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.90 + 8.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.74 + 3.01i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.33 - 12.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.69 + 8.13i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.889 + 1.54i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.92 + 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−10.21210794905088949558567179622, −9.428299740166385690691100963504, −9.231188529448878037025636362510, −7.74100752759276189601839813131, −6.46156407031159995931443052513, −5.57902917481545513561798381948, −4.76864946720457347648302997217, −4.07893297390514749577168357597, −3.29603508796149165184346272788, −0.61757441131443800109988068587, 1.21401756038726033894393629545, 2.72764148033239938425832327546, 4.02443960501742115813445897447, 5.29097881563285495549911566949, 6.10481736254822238027626752557, 6.57458921213352266680712142762, 7.65867728508925520402844189788, 8.777273791223622777636664345104, 9.451153807291200739354599395026, 10.50801227986753015120016066454

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