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Properties

Label 2-731-43.6-c1-0-33
Degree $2$
Conductor $731$
Sign $0.284 + 0.958i$
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  − 1.99·2-s + (0.176 − 0.306i)3-s + 1.96·4-s + (−1.32 + 2.29i)5-s + (−0.352 + 0.609i)6-s + (−1.66 − 2.88i)7-s + 0.0645·8-s + (1.43 + 2.48i)9-s + (2.64 − 4.57i)10-s + 0.00668·11-s + (0.347 − 0.602i)12-s + (−2.08 − 3.60i)13-s + (3.31 + 5.74i)14-s + (0.468 + 0.811i)15-s − 4.06·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 1.40·2-s + (0.102 − 0.176i)3-s + 0.983·4-s + (−0.592 + 1.02i)5-s + (−0.143 + 0.249i)6-s + (−0.629 − 1.08i)7-s + 0.0228·8-s + (0.479 + 0.829i)9-s + (0.835 − 1.44i)10-s + 0.00201·11-s + (0.100 − 0.173i)12-s + (−0.577 − 1.00i)13-s + (0.885 + 1.53i)14-s + (0.121 + 0.209i)15-s − 1.01·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.284 + 0.958i$
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: \((2,\ 731,\ (\ :1/2),\ 0.284 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339965 - 0.253794i\)
\(L(\frac12)\) \(\approx\) \(0.339965 - 0.253794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.109 - 6.55i)T \)
good2 \( 1 + 1.99T + 2T^{2} \)
3 \( 1 + (-0.176 + 0.306i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.32 - 2.29i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.66 + 2.88i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.00668T + 11T^{2} \)
13 \( 1 + (2.08 + 3.60i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (3.12 - 5.41i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.88 + 3.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.53 + 2.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.02 + 5.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.28 + 5.68i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.31T + 41T^{2} \)
47 \( 1 + 8.80T + 47T^{2} \)
53 \( 1 + (-5.34 + 9.26i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.15T + 59T^{2} \)
61 \( 1 + (7.67 + 13.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.48 + 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3.46 - 5.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.98 + 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.616 - 1.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.22 + 14.2i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.65 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−10.22662510146695203448940607155, −9.645319513517330592934295659630, −8.135003641121149537271974711113, −7.80372129318048352732079172009, −7.12941105982507642864293593058, −6.29671372311390130941828284193, −4.55362765440197989840682390017, −3.44460137626470341785004673379, −2.13131454907187363868727285144, −0.41684965766728224870518905412, 1.08764576343254726593305068900, 2.61176989927160535562080299352, 4.17706507086199715637407371007, 5.07185644396327985786311880915, 6.57025932596601668582430308274, 7.21493952156229460036474804347, 8.453951959383452569062159463603, 8.984768404769156540378408583576, 9.345768795041080617151465151029, 10.16030556326409406693294702570

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