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Properties

Label 2-731-43.35-c1-0-10
Degree $2$
Conductor $731$
Sign $-0.718 + 0.695i$
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  + (−0.265 + 1.16i)2-s + (0.558 + 2.44i)3-s + (0.522 + 0.251i)4-s + (0.802 − 1.00i)5-s − 2.99·6-s − 4.26·7-s + (−1.91 + 2.40i)8-s + (−2.97 + 1.43i)9-s + (0.956 + 1.19i)10-s + (−0.515 + 0.248i)11-s + (−0.324 + 1.41i)12-s + (−0.401 + 0.504i)13-s + (1.12 − 4.95i)14-s + (2.91 + 1.40i)15-s + (−1.56 − 1.95i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.187 + 0.821i)2-s + (0.322 + 1.41i)3-s + (0.261 + 0.125i)4-s + (0.359 − 0.450i)5-s − 1.22·6-s − 1.61·7-s + (−0.677 + 0.849i)8-s + (−0.991 + 0.477i)9-s + (0.302 + 0.379i)10-s + (−0.155 + 0.0748i)11-s + (−0.0935 + 0.409i)12-s + (−0.111 + 0.139i)13-s + (0.301 − 1.32i)14-s + (0.752 + 0.362i)15-s + (−0.390 − 0.489i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.718 + 0.695i$
Analytic conductor: \(5.83706\)
Root analytic conductor: \(2.41600\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 731,\ (\ :1/2),\ -0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.391557 - 0.968389i\)
\(L(\frac12)\) \(\approx\) \(0.391557 - 0.968389i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (4.56 - 4.71i)T \)
good2 \( 1 + (0.265 - 1.16i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.558 - 2.44i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.802 + 1.00i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 4.26T + 7T^{2} \)
11 \( 1 + (0.515 - 0.248i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.401 - 0.504i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (4.37 + 2.10i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (1.52 - 0.734i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.0857 - 0.375i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.414 + 1.81i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 3.73T + 37T^{2} \)
41 \( 1 + (0.723 - 3.17i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-4.70 - 2.26i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-4.92 - 6.17i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (3.11 + 3.90i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-2.68 - 11.7i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-8.64 - 4.16i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (7.62 + 3.67i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-1.88 + 2.36i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 3.81T + 79T^{2} \)
83 \( 1 + (-0.0628 - 0.275i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (3.74 + 16.4i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-6.94 + 3.34i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.57192115915943369378120616958, −9.850733258304005061360580061600, −9.168763230042200954902244077874, −8.630720983443332907078680719080, −7.41911100422743327616463641358, −6.42021637096445214832015729754, −5.73285912843392386131155529349, −4.62103103190488983196434843852, −3.52045733671344342353951149549, −2.59864942160497962582698172624, 0.49870485619320573855908578344, 2.05339188743739812870998482529, 2.70990873231086114888126264228, 3.67572308893983115931872430769, 5.84087491926156921631284032848, 6.58685390890967468700246228727, 6.91214428558448311941093954277, 8.118479321336033032719406877921, 9.159362390864036458361744190560, 10.09968053117267224958437011121

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