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Properties

Label 2-731-731.16-c1-0-58
Degree $2$
Conductor $731$
Sign $-0.617 + 0.786i$
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  + (−0.473 − 2.07i)2-s + (3.20 + 0.730i)3-s + (−2.28 + 1.09i)4-s + (0.606 − 0.483i)5-s − 6.98i·6-s − 4.01i·7-s + (0.704 + 0.883i)8-s + (7.01 + 3.37i)9-s + (−1.29 − 1.02i)10-s + (0.0655 − 0.136i)11-s + (−8.10 + 1.84i)12-s + (−2.82 − 3.53i)13-s + (−8.33 + 1.90i)14-s + (2.29 − 1.10i)15-s + (−1.65 + 2.07i)16-s + (−1.81 + 3.70i)17-s + ⋯
L(s)  = 1  + (−0.334 − 1.46i)2-s + (1.84 + 0.421i)3-s + (−1.14 + 0.549i)4-s + (0.271 − 0.216i)5-s − 2.85i·6-s − 1.51i·7-s + (0.249 + 0.312i)8-s + (2.33 + 1.12i)9-s + (−0.408 − 0.325i)10-s + (0.0197 − 0.0410i)11-s + (−2.33 + 0.533i)12-s + (−0.782 − 0.981i)13-s + (−2.22 + 0.508i)14-s + (0.592 − 0.285i)15-s + (−0.414 + 0.519i)16-s + (−0.440 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 + 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.617 + 0.786i$
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: \((2,\ 731,\ (\ :1/2),\ -0.617 + 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02618 - 2.11167i\)
\(L(\frac12)\) \(\approx\) \(1.02618 - 2.11167i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.81 - 3.70i)T \)
43 \( 1 + (5.20 - 3.99i)T \)
good2 \( 1 + (0.473 + 2.07i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-3.20 - 0.730i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.606 + 0.483i)T + (1.11 - 4.87i)T^{2} \)
7 \( 1 + 4.01iT - 7T^{2} \)
11 \( 1 + (-0.0655 + 0.136i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (2.82 + 3.53i)T + (-2.89 + 12.6i)T^{2} \)
19 \( 1 + (-5.48 + 2.64i)T + (11.8 - 14.8i)T^{2} \)
23 \( 1 + (2.64 - 5.48i)T + (-14.3 - 17.9i)T^{2} \)
29 \( 1 + (-3.86 + 0.881i)T + (26.1 - 12.5i)T^{2} \)
31 \( 1 + (5.46 - 1.24i)T + (27.9 - 13.4i)T^{2} \)
37 \( 1 + 1.77iT - 37T^{2} \)
41 \( 1 + (0.498 - 0.113i)T + (36.9 - 17.7i)T^{2} \)
47 \( 1 + (-6.46 + 3.11i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.96 + 4.97i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + (-8.70 + 10.9i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-6.39 - 1.45i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (7.90 - 3.80i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-3.49 - 7.25i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (0.479 - 0.382i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + 5.60iT - 79T^{2} \)
83 \( 1 + (-2.74 + 12.0i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (0.803 - 3.51i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (7.83 - 16.2i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.02184628729166127887733303976, −9.547905346151708384484966026500, −8.692058292314161330712355527834, −7.77082052717602343452897291853, −7.13011556861716135347104465812, −4.99515977376758344453025485172, −3.79108160837057042880287350975, −3.45858911014511380378306867519, −2.30652224918906616352475239031, −1.22077757913068802737422980624, 2.17758368282997241454446833951, 2.78950081018175184310255082329, 4.43000629598099350746940418181, 5.62883736154895875700580346577, 6.70976636014566057154378723393, 7.26105632986906774147060194863, 8.189658927794356413844862550637, 8.760886677125735079892970177298, 9.350274864559441684536442366533, 9.922874156992444418504194085090

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