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Properties

Label 2-731-731.208-c1-0-19
Degree $2$
Conductor $731$
Sign $-0.587 + 0.809i$
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  − 2.67i·2-s + (−2.11 − 0.565i)3-s − 5.15·4-s + (2.13 + 0.573i)5-s + (−1.51 + 5.64i)6-s + (0.737 − 0.197i)7-s + 8.43i·8-s + (1.53 + 0.887i)9-s + (1.53 − 5.72i)10-s + (4.20 + 4.20i)11-s + (10.8 + 2.91i)12-s + (2.31 − 4.00i)13-s + (−0.528 − 1.97i)14-s + (−4.19 − 2.41i)15-s + 12.2·16-s + (1.61 + 3.79i)17-s + ⋯
L(s)  = 1  − 1.89i·2-s + (−1.21 − 0.326i)3-s − 2.57·4-s + (0.956 + 0.256i)5-s + (−0.617 + 2.30i)6-s + (0.278 − 0.0747i)7-s + 2.98i·8-s + (0.512 + 0.295i)9-s + (0.484 − 1.80i)10-s + (1.26 + 1.26i)11-s + (3.14 + 0.841i)12-s + (0.641 − 1.11i)13-s + (−0.141 − 0.527i)14-s + (−1.08 − 0.624i)15-s + 3.06·16-s + (0.390 + 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.587 + 0.809i$
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: \((2,\ 731,\ (\ :1/2),\ -0.587 + 0.809i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.504241 - 0.988807i\)
\(L(\frac12)\) \(\approx\) \(0.504241 - 0.988807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.61 - 3.79i)T \)
43 \( 1 + (-2.18 + 6.18i)T \)
good2 \( 1 + 2.67iT - 2T^{2} \)
3 \( 1 + (2.11 + 0.565i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-2.13 - 0.573i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.737 + 0.197i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.20 - 4.20i)T + 11iT^{2} \)
13 \( 1 + (-2.31 + 4.00i)T + (-6.5 - 11.2i)T^{2} \)
19 \( 1 + (-3.64 + 2.10i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.79 - 6.70i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.860 + 3.21i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.46 - 0.927i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (5.62 + 1.50i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-7.28 - 7.28i)T + 41iT^{2} \)
47 \( 1 - 3.34T + 47T^{2} \)
53 \( 1 + (-3.31 + 1.91i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 2.62iT - 59T^{2} \)
61 \( 1 + (8.13 - 2.17i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.09 + 1.89i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.02 - 7.56i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.0701 + 0.261i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.92 + 1.31i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-9.48 + 5.47i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (8.82 + 15.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.04 + 6.04i)T - 97iT^{2} \)
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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

−10.20947687669841891653907483956, −9.781502519366289279674430131728, −8.883028690771765400848379828052, −7.53745691328112202681592176111, −6.12161538655463445001766315296, −5.49176152757036479888796876542, −4.42264350565729871140272212310, −3.31588414762819907100458844699, −1.85050686832634275276349171542, −1.09674111859423917209704350246, 0.955806177829343586864324946618, 3.87914281650487996129082776102, 4.89261118773293473386521413669, 5.61106856092519109237209257864, 6.25684968449250380832960687630, 6.68495075637088171112439767883, 8.010385779521152743995434972639, 9.028659135382429608147790354860, 9.357514233822393275504707163350, 10.51459410965433360897013891408

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