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Properties

Label 2-731-43.35-c1-0-33
Degree $2$
Conductor $731$
Sign $0.617 - 0.786i$
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  + (0.0448 − 0.196i)2-s + (0.586 + 2.56i)3-s + (1.76 + 0.850i)4-s + (2.40 − 3.01i)5-s + 0.530·6-s + 0.970·7-s + (0.497 − 0.623i)8-s + (−3.54 + 1.70i)9-s + (−0.484 − 0.607i)10-s + (−3.37 + 1.62i)11-s + (−1.14 + 5.03i)12-s + (−0.449 + 0.563i)13-s + (0.0435 − 0.190i)14-s + (9.15 + 4.40i)15-s + (2.34 + 2.93i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (0.0317 − 0.138i)2-s + (0.338 + 1.48i)3-s + (0.882 + 0.425i)4-s + (1.07 − 1.34i)5-s + 0.216·6-s + 0.366·7-s + (0.175 − 0.220i)8-s + (−1.18 + 0.569i)9-s + (−0.153 − 0.192i)10-s + (−1.01 + 0.489i)11-s + (−0.331 + 1.45i)12-s + (−0.124 + 0.156i)13-s + (0.0116 − 0.0509i)14-s + (2.36 + 1.13i)15-s + (0.585 + 0.734i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.617 - 0.786i$
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: \((2,\ 731,\ (\ :1/2),\ 0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.23796 + 1.08796i\)
\(L(\frac12)\) \(\approx\) \(2.23796 + 1.08796i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.87 + 5.28i)T \)
good2 \( 1 + (-0.0448 + 0.196i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.586 - 2.56i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-2.40 + 3.01i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 - 0.970T + 7T^{2} \)
11 \( 1 + (3.37 - 1.62i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.449 - 0.563i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (1.04 + 0.501i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-3.59 + 1.73i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (0.196 - 0.859i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (1.06 - 4.65i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 + (-1.72 + 7.53i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-4.11 - 1.97i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (-2.50 - 3.14i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (3.94 + 4.95i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (0.944 + 4.13i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (5.08 + 2.44i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-9.41 - 4.53i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.37 - 1.71i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (0.397 + 1.74i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (2.67 + 11.7i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (16.3 - 7.87i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.47519160781540259774111737809, −9.717073819242892140468001225323, −8.893854010956336691309184389975, −8.296421929068522581576043035008, −7.08197809803686902771685906337, −5.65397177605684613932535039405, −5.02580800069242533564679125260, −4.19183216705192390937866488083, −2.87122122008117984997285365882, −1.78566423794455929967969776299, 1.44886124420243262012034476087, 2.46585443654409659035885491480, 2.93542724545751241286638487667, 5.40623138483325786193284300044, 6.08044522847070708273684379841, 6.79978534679425115291761017265, 7.46487390085768308299132020729, 8.105960237656623222217028680199, 9.532648445076400509248936165408, 10.48032752097633825755493742175

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