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Properties

Label 2-731-731.208-c1-0-45
Degree $2$
Conductor $731$
Sign $0.781 + 0.624i$
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  − 1.24i·2-s + (2.55 + 0.683i)3-s + 0.459·4-s + (1.75 + 0.470i)5-s + (0.848 − 3.16i)6-s + (−1.74 + 0.467i)7-s − 3.05i·8-s + (3.44 + 1.98i)9-s + (0.584 − 2.18i)10-s + (1.36 + 1.36i)11-s + (1.17 + 0.313i)12-s + (−1.50 + 2.60i)13-s + (0.580 + 2.16i)14-s + (4.15 + 2.40i)15-s − 2.87·16-s + (1.26 − 3.92i)17-s + ⋯
L(s)  = 1  − 0.877i·2-s + (1.47 + 0.394i)3-s + 0.229·4-s + (0.785 + 0.210i)5-s + (0.346 − 1.29i)6-s + (−0.659 + 0.176i)7-s − 1.07i·8-s + (1.14 + 0.662i)9-s + (0.184 − 0.689i)10-s + (0.411 + 0.411i)11-s + (0.338 + 0.0906i)12-s + (−0.417 + 0.722i)13-s + (0.155 + 0.579i)14-s + (1.07 + 0.619i)15-s − 0.717·16-s + (0.307 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.781 + 0.624i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.781 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.79304 - 0.978399i\)
\(L(\frac12)\) \(\approx\) \(2.79304 - 0.978399i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.26 + 3.92i)T \)
43 \( 1 + (-6.20 + 2.11i)T \)
good2 \( 1 + 1.24iT - 2T^{2} \)
3 \( 1 + (-2.55 - 0.683i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.75 - 0.470i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.74 - 0.467i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.36 - 1.36i)T + 11iT^{2} \)
13 \( 1 + (1.50 - 2.60i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-0.780 + 0.450i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.978 - 3.65i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.0319 - 0.119i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (9.88 + 2.64i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-3.88 - 1.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.90 - 2.90i)T + 41iT^{2} \)
47 \( 1 - 0.665T + 47T^{2} \)
53 \( 1 + (9.31 - 5.38i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.34iT - 59T^{2} \)
61 \( 1 + (-9.86 + 2.64i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.292 - 0.507i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.34 + 8.73i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.204 - 0.763i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (10.3 - 2.76i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.42 + 1.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.76 + 3.06i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.19 + 2.19i)T - 97iT^{2} \)
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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

−9.934037273830496876008284475822, −9.435982773045086196430664260814, −9.270859945702673813313002972428, −7.70123842729573985332443312349, −6.98586574152915933053651935283, −5.89237805401288834720861528245, −4.31199119284303184756776429766, −3.37374115964231725123545669997, −2.56562497211607447260277084083, −1.79616363435626823363773724097, 1.74875572695164186908643292117, 2.75399659152618377974733906136, 3.77014272958008126273223944778, 5.46311736273540599244600662765, 6.19620327003181759042779661672, 7.13029387234834422395003389393, 7.86534980881981676765916921726, 8.590039643639459417562688328581, 9.348949171027603850174008017611, 10.15809512412847003028539678722

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