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Properties

Label 2-731-17.4-c1-0-57
Degree $2$
Conductor $731$
Sign $-0.615 + 0.788i$
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  − i·2-s + (2 − 2i)3-s + 4-s + (−2 − 2i)6-s − 3i·8-s − 5i·9-s + (−1 − i)11-s + (2 − 2i)12-s + 2·13-s − 16-s + (−4 + i)17-s − 5·18-s + 8i·19-s + (−1 + i)22-s + (−1 − i)23-s + (−6 − 6i)24-s + ⋯
L(s)  = 1  − 0.707i·2-s + (1.15 − 1.15i)3-s + 0.5·4-s + (−0.816 − 0.816i)6-s − 1.06i·8-s − 1.66i·9-s + (−0.301 − 0.301i)11-s + (0.577 − 0.577i)12-s + 0.554·13-s − 0.250·16-s + (−0.970 + 0.242i)17-s − 1.17·18-s + 1.83i·19-s + (−0.213 + 0.213i)22-s + (−0.208 − 0.208i)23-s + (−1.22 − 1.22i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.615 + 0.788i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.615 + 0.788i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12244 - 2.30042i\)
\(L(\frac12)\) \(\approx\) \(1.12244 - 2.30042i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (4 - i)T \)
43 \( 1 - iT \)
good2 \( 1 + iT - 2T^{2} \)
3 \( 1 + (-2 + 2i)T - 3iT^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + (1 + i)T + 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 - 8iT - 19T^{2} \)
23 \( 1 + (1 + i)T + 23iT^{2} \)
29 \( 1 + (-6 + 6i)T - 29iT^{2} \)
31 \( 1 + (-1 + i)T - 31iT^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 + (-1 - i)T + 41iT^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (6 + 6i)T + 61iT^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (4 - 4i)T - 71iT^{2} \)
73 \( 1 + (-6 + 6i)T - 73iT^{2} \)
79 \( 1 + (-7 - 7i)T + 79iT^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−10.16783068376124761506912308083, −9.157415864536252235318450975672, −8.232860858247973381062938477576, −7.67574378349246143757087391030, −6.65366364730885702834720601995, −5.98565275205293651201228046299, −4.02391036294166465288780488830, −3.11478129124025493411316977348, −2.18391306438087379011073993523, −1.28023845653177343995193215651, 2.30334397759167169838240475932, 3.07378971741742217517660556081, 4.42070198147615999439370557529, 5.10361443357043953192677230414, 6.47470835122132536280224377347, 7.23887016971579815391748163711, 8.360892683677072043317692309069, 8.757951661733755559047528531306, 9.660315414897447720112299911427, 10.68967811705408164721969778838

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