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Properties

Label 2-731-731.135-c1-0-37
Degree $2$
Conductor $731$
Sign $-0.947 - 0.320i$
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  − 2.04·2-s + (−1.14 − 0.660i)3-s + 2.20·4-s + (−1.07 − 0.619i)5-s + (2.34 + 1.35i)6-s + (0.325 − 0.188i)7-s − 0.411·8-s + (−0.626 − 1.08i)9-s + (2.19 + 1.26i)10-s + 3.21i·11-s + (−2.51 − 1.45i)12-s + (−1.51 − 2.63i)13-s + (−0.667 + 0.385i)14-s + (0.818 + 1.41i)15-s − 3.55·16-s + (1.69 − 3.75i)17-s + ⋯
L(s)  = 1  − 1.44·2-s + (−0.660 − 0.381i)3-s + 1.10·4-s + (−0.479 − 0.276i)5-s + (0.957 + 0.552i)6-s + (0.123 − 0.0710i)7-s − 0.145·8-s + (−0.208 − 0.361i)9-s + (0.695 + 0.401i)10-s + 0.969i·11-s + (−0.727 − 0.419i)12-s + (−0.421 − 0.729i)13-s + (−0.178 + 0.103i)14-s + (0.211 + 0.366i)15-s − 0.889·16-s + (0.411 − 0.911i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.947 - 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0205032 + 0.124771i\)
\(L(\frac12)\) \(\approx\) \(0.0205032 + 0.124771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.69 + 3.75i)T \)
43 \( 1 + (6.23 - 2.04i)T \)
good2 \( 1 + 2.04T + 2T^{2} \)
3 \( 1 + (1.14 + 0.660i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.07 + 0.619i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.325 + 0.188i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.21iT - 11T^{2} \)
13 \( 1 + (1.51 + 2.63i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + (-2.17 + 3.76i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.19 - 3.57i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.12 + 1.22i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.76 + 2.75i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0633 + 0.0365i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.40iT - 41T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + (0.0597 - 0.103i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.51T + 59T^{2} \)
61 \( 1 + (9.01 - 5.20i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.84 + 10.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.83 - 3.94i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (8.42 - 4.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.72 - 2.15i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.85 - 15.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.49 - 4.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 13.4iT - 97T^{2} \)
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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

−9.647672784913081126363162034976, −9.263142244388707285938320073203, −8.161804396488676319916742506783, −7.33528071429744314074747452502, −6.93614049600445174319726402117, −5.50271220214535102897750676393, −4.58180531515219452171754704590, −2.87548307352137818880161067924, −1.24313786079383942604253540984, −0.13287109926531372845760141664, 1.56370366480131681196642197467, 3.21520822971375089714787293769, 4.58351543053732633300213293208, 5.65137788286154655496595733239, 6.71818330794092185879322701834, 7.65679763575454778384106074041, 8.382991103571847364593555467797, 9.096869738640303137011978345064, 10.14455346035968966431531501694, 10.68745738660013257173652426285

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