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Properties

Label 2-731-43.35-c1-0-0
Degree $2$
Conductor $731$
Sign $-0.286 + 0.958i$
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  + (−0.598 + 2.62i)2-s + (−0.0443 − 0.194i)3-s + (−4.72 − 2.27i)4-s + (2.45 − 3.07i)5-s + 0.536·6-s − 2.50·7-s + (5.44 − 6.83i)8-s + (2.66 − 1.28i)9-s + (6.60 + 8.27i)10-s + (−5.75 + 2.77i)11-s + (−0.232 + 1.01i)12-s + (−3.10 + 3.89i)13-s + (1.50 − 6.57i)14-s + (−0.705 − 0.339i)15-s + (8.12 + 10.1i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.423 + 1.85i)2-s + (−0.0255 − 0.112i)3-s + (−2.36 − 1.13i)4-s + (1.09 − 1.37i)5-s + 0.218·6-s − 0.947·7-s + (1.92 − 2.41i)8-s + (0.889 − 0.428i)9-s + (2.08 + 2.61i)10-s + (−1.73 + 0.835i)11-s + (−0.0671 + 0.294i)12-s + (−0.861 + 1.08i)13-s + (0.401 − 1.75i)14-s + (−0.182 − 0.0877i)15-s + (2.03 + 2.54i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.286 + 0.958i$
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: \((2,\ 731,\ (\ :1/2),\ -0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0518923 - 0.0696805i\)
\(L(\frac12)\) \(\approx\) \(0.0518923 - 0.0696805i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 6.34i)T \)
good2 \( 1 + (0.598 - 2.62i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (0.0443 + 0.194i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (-2.45 + 3.07i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + 2.50T + 7T^{2} \)
11 \( 1 + (5.75 - 2.77i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.10 - 3.89i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (1.45 + 0.702i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (7.11 - 3.42i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (1.48 - 6.49i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (-0.976 + 4.27i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + 1.02T + 37T^{2} \)
41 \( 1 + (0.218 - 0.955i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (1.53 + 0.741i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (4.35 + 5.45i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (0.00131 + 0.00165i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-0.788 - 3.45i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (0.151 + 0.0729i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-4.06 - 1.95i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (1.65 - 2.07i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 + (0.921 + 4.03i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.964 + 4.22i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (4.93 - 2.37i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.06071088400151845082100674624, −9.808873201565360598902415779631, −9.311794561060118799698166836876, −8.284000125868013348677543952083, −7.44880814915931409019958009406, −6.64075051703931062185267573472, −5.81005333339595656913184171166, −5.00695064232909360324760018826, −4.33758051537758359848512508690, −1.79048910553180272248173814691, 0.04869926630666191244286517868, 2.20746358664193505793598523413, 2.71963096191262713425671636274, 3.55507308518351522842498478413, 5.04023767832294750509309108274, 6.04734660257196972578697340593, 7.46937674518683559506663965342, 8.259819464997252176037947368094, 9.727061458548375891484667637591, 10.02954467015396560927620636187

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