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Properties

Label 2-731-17.2-c1-0-36
Degree $2$
Conductor $731$
Sign $0.693 + 0.720i$
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  + (0.639 + 0.639i)2-s + (−0.916 + 0.379i)3-s − 1.18i·4-s + (0.199 + 0.482i)5-s + (−0.829 − 0.343i)6-s + (0.462 − 1.11i)7-s + (2.03 − 2.03i)8-s + (−1.42 + 1.42i)9-s + (−0.180 + 0.436i)10-s + (1.30 + 0.540i)11-s + (0.448 + 1.08i)12-s − 6.44i·13-s + (1.01 − 0.418i)14-s + (−0.366 − 0.366i)15-s + 0.239·16-s + (−2.99 − 2.83i)17-s + ⋯
L(s)  = 1  + (0.452 + 0.452i)2-s + (−0.529 + 0.219i)3-s − 0.590i·4-s + (0.0893 + 0.215i)5-s + (−0.338 − 0.140i)6-s + (0.174 − 0.422i)7-s + (0.719 − 0.719i)8-s + (−0.475 + 0.475i)9-s + (−0.0571 + 0.137i)10-s + (0.393 + 0.162i)11-s + (0.129 + 0.312i)12-s − 1.78i·13-s + (0.270 − 0.111i)14-s + (−0.0945 − 0.0945i)15-s + 0.0599·16-s + (−0.726 − 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.693 + 0.720i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38975 - 0.590811i\)
\(L(\frac12)\) \(\approx\) \(1.38975 - 0.590811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.99 + 2.83i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (-0.639 - 0.639i)T + 2iT^{2} \)
3 \( 1 + (0.916 - 0.379i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (-0.199 - 0.482i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-0.462 + 1.11i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.30 - 0.540i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 6.44iT - 13T^{2} \)
19 \( 1 + (-3.13 - 3.13i)T + 19iT^{2} \)
23 \( 1 + (5.39 + 2.23i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (-1.18 - 2.86i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-5.43 + 2.25i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (-6.99 + 2.89i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.27 + 3.06i)T + (-28.9 - 28.9i)T^{2} \)
47 \( 1 + 5.93iT - 47T^{2} \)
53 \( 1 + (7.37 + 7.37i)T + 53iT^{2} \)
59 \( 1 + (-3.01 + 3.01i)T - 59iT^{2} \)
61 \( 1 + (5.47 - 13.2i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 8.65T + 67T^{2} \)
71 \( 1 + (7.66 - 3.17i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (-3.55 - 8.57i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-2.04 - 0.848i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (1.34 + 1.34i)T + 83iT^{2} \)
89 \( 1 - 5.67iT - 89T^{2} \)
97 \( 1 + (0.981 + 2.36i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.34827238052661213244812291456, −9.775817038948278582744239203748, −8.364395485522130660091702225885, −7.54984685298962651231718749828, −6.49950338453423055177510082611, −5.74587975192863470853229024687, −5.03756085712606152634476627428, −4.11933827083343227086883567218, −2.59949471337649554294129993770, −0.75378142468466572614417272079, 1.63300002815619780125755789464, 2.89151367363156536886527135147, 4.10378131694868941644376656968, 4.86196000765774084726005968289, 6.11577504208245779787712258855, 6.78431853307707903263680207161, 7.971994761762726560378119024577, 8.894621678653464398163192199977, 9.469409876467414458424240937099, 11.00218340126999025076522665527

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