Properties

Label 2-731-43.6-c1-0-1
Degree $2$
Conductor $731$
Sign $-0.948 + 0.316i$
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  − 0.317·2-s + (−0.924 + 1.60i)3-s − 1.89·4-s + (1.30 − 2.26i)5-s + (0.293 − 0.508i)6-s + (0.730 + 1.26i)7-s + 1.23·8-s + (−0.210 − 0.364i)9-s + (−0.415 + 0.719i)10-s − 3.65·11-s + (1.75 − 3.04i)12-s + (−1.01 − 1.75i)13-s + (−0.231 − 0.401i)14-s + (2.41 + 4.18i)15-s + 3.40·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 0.224·2-s + (−0.533 + 0.924i)3-s − 0.949·4-s + (0.584 − 1.01i)5-s + (0.119 − 0.207i)6-s + (0.276 + 0.478i)7-s + 0.437·8-s + (−0.0701 − 0.121i)9-s + (−0.131 + 0.227i)10-s − 1.10·11-s + (0.506 − 0.878i)12-s + (−0.281 − 0.486i)13-s + (−0.0619 − 0.107i)14-s + (0.624 + 1.08i)15-s + 0.851·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.948 + 0.316i$
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: \((2,\ 731,\ (\ :1/2),\ -0.948 + 0.316i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0178253 - 0.109695i\)
\(L(\frac12)\) \(\approx\) \(0.0178253 - 0.109695i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.55 - 0.113i)T \)
good2 \( 1 + 0.317T + 2T^{2} \)
3 \( 1 + (0.924 - 1.60i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.30 + 2.26i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.730 - 1.26i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + (1.01 + 1.75i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (2.93 - 5.07i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.46 - 2.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.83 + 6.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.86 + 4.96i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.832 - 1.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + (2.10 - 3.64i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 3.29T + 59T^{2} \)
61 \( 1 + (4.13 + 7.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.90 + 6.77i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.20 - 2.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.29 - 2.23i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.697 + 1.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.09 - 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.42 - 5.93i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8.27T + 97T^{2} \)
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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

−10.45042991340002175300410559544, −9.958063994407507701068277317859, −9.404711040244689572022287768981, −8.252412744205317525752931374058, −7.960160251249867394622340215298, −5.92188506844269319836360348979, −5.26667530156645555897843804298, −4.80989131532317099236546519882, −3.74257010719885468535395822367, −1.85172455393364980459204766693, 0.06645189623926036339259581703, 1.71186317036540870913148434001, 3.09037380171755199221418641055, 4.58802439736202351315619868397, 5.45244985734398401842052135211, 6.67004916202410886374173068321, 7.08936759945238552789249094316, 8.101326665532920373545733055566, 9.054618595184241763923233109940, 10.15665734670100375901241705571

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