Properties

Label 2-731-43.6-c1-0-17
Degree $2$
Conductor $731$
Sign $0.967 - 0.254i$
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  − 2.13·2-s + (1.01 − 1.76i)3-s + 2.56·4-s + (−1.08 + 1.88i)5-s + (−2.17 + 3.77i)6-s + (0.371 + 0.642i)7-s − 1.21·8-s + (−0.575 − 0.996i)9-s + (2.32 − 4.02i)10-s + 0.410·11-s + (2.61 − 4.52i)12-s + (−1.10 − 1.91i)13-s + (−0.792 − 1.37i)14-s + (2.21 + 3.83i)15-s − 2.54·16-s + (0.5 + 0.866i)17-s + ⋯
L(s)  = 1  − 1.51·2-s + (0.588 − 1.01i)3-s + 1.28·4-s + (−0.485 + 0.841i)5-s + (−0.888 + 1.53i)6-s + (0.140 + 0.242i)7-s − 0.428·8-s + (−0.191 − 0.332i)9-s + (0.734 − 1.27i)10-s + 0.123·11-s + (0.754 − 1.30i)12-s + (−0.307 − 0.531i)13-s + (−0.211 − 0.367i)14-s + (0.571 + 0.989i)15-s − 0.635·16-s + (0.121 + 0.210i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.967 - 0.254i$
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: \((2,\ 731,\ (\ :1/2),\ 0.967 - 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767576 + 0.0994152i\)
\(L(\frac12)\) \(\approx\) \(0.767576 + 0.0994152i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.55 - 0.310i)T \)
good2 \( 1 + 2.13T + 2T^{2} \)
3 \( 1 + (-1.01 + 1.76i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.08 - 1.88i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.371 - 0.642i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 0.410T + 11T^{2} \)
13 \( 1 + (1.10 + 1.91i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-3.07 + 5.32i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.88 - 6.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.16 - 7.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.08 - 5.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.44 - 5.97i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.165T + 41T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 + (-3.48 + 6.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.68T + 59T^{2} \)
61 \( 1 + (5.48 + 9.50i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.52 + 6.11i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.52 - 11.2i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.63 - 4.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.28 - 3.94i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.21 - 5.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.9T + 97T^{2} \)
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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

−10.36869159154888939856867265139, −9.365474050158481495644893185342, −8.594713140786708657218325168944, −7.84844867043820438788190191310, −7.18996661390584017145167702385, −6.79855792960117747119480084808, −5.19746924361335580643113499668, −3.34710714931922403065097341945, −2.35336035283431010728862017794, −1.16006826720581262957949762946, 0.74438667001373731688709171903, 2.35191595940169296753463153320, 4.03930527411279478363640721437, 4.49733991299611760978694708182, 6.06171685599220936692403450630, 7.47349943618599476295095703669, 8.002674504565114097252832933932, 8.963208469458432135983554090308, 9.234818009616860770997839768365, 10.21772784061531057945919101135

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