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Properties

Label 2-731-43.35-c1-0-9
Degree $2$
Conductor $731$
Sign $0.725 + 0.688i$
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  + (−0.534 + 2.34i)2-s + (0.544 + 2.38i)3-s + (−3.39 − 1.63i)4-s + (−2.43 + 3.04i)5-s − 5.88·6-s + 1.85·7-s + (2.65 − 3.32i)8-s + (−2.69 + 1.29i)9-s + (−5.84 − 7.32i)10-s + (0.554 − 0.267i)11-s + (2.05 − 8.99i)12-s + (1.74 − 2.19i)13-s + (−0.990 + 4.34i)14-s + (−8.60 − 4.14i)15-s + (1.66 + 2.09i)16-s + (0.623 + 0.781i)17-s + ⋯
L(s)  = 1  + (−0.377 + 1.65i)2-s + (0.314 + 1.37i)3-s + (−1.69 − 0.817i)4-s + (−1.08 + 1.36i)5-s − 2.40·6-s + 0.700·7-s + (0.937 − 1.17i)8-s + (−0.898 + 0.432i)9-s + (−1.84 − 2.31i)10-s + (0.167 − 0.0805i)11-s + (0.592 − 2.59i)12-s + (0.484 − 0.607i)13-s + (−0.264 + 1.16i)14-s + (−2.22 − 1.06i)15-s + (0.416 + 0.522i)16-s + (0.151 + 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $0.725 + 0.688i$
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: \((2,\ 731,\ (\ :1/2),\ 0.725 + 0.688i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.677534 - 0.270424i\)
\(L(\frac12)\) \(\approx\) \(0.677534 - 0.270424i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.90 - 4.35i)T \)
good2 \( 1 + (0.534 - 2.34i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-0.544 - 2.38i)T + (-2.70 + 1.30i)T^{2} \)
5 \( 1 + (2.43 - 3.04i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 - 1.85T + 7T^{2} \)
11 \( 1 + (-0.554 + 0.267i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-1.74 + 2.19i)T + (-2.89 - 12.6i)T^{2} \)
19 \( 1 + (6.60 + 3.18i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (4.07 - 1.96i)T + (14.3 - 17.9i)T^{2} \)
29 \( 1 + (1.53 - 6.74i)T + (-26.1 - 12.5i)T^{2} \)
31 \( 1 + (0.661 - 2.89i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 + (2.03 - 8.90i)T + (-36.9 - 17.7i)T^{2} \)
47 \( 1 + (-1.64 - 0.789i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (6.17 + 7.74i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + (-0.949 - 1.19i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (0.568 + 2.48i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-4.32 - 2.08i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (11.5 + 5.55i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (3.47 - 4.35i)T + (-16.2 - 71.1i)T^{2} \)
79 \( 1 - 6.85T + 79T^{2} \)
83 \( 1 + (-0.353 - 1.54i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (2.31 + 10.1i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (2.88 - 1.38i)T + (60.4 - 75.8i)T^{2} \)
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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

−10.90500503392129259275319925230, −10.16790707937972894635843657780, −9.190853996653550314911538386231, −8.277643967145027756394356321482, −7.87092631212438613726043570337, −6.84197738170006954910490907529, −6.04600617634647059921058234510, −4.79644569034619491549871817994, −4.10959602269956588985784379201, −3.07106720174502367482600114845, 0.43499730072356171869367926734, 1.50818217838983811371278077422, 2.26811005364698174296867217617, 4.01662866054322618453479712828, 4.37855938759854570187620941459, 6.07693508475616114236848674146, 7.56429615535865045495726060894, 8.232724593158646531193404381581, 8.633914297123182703391029812960, 9.545695878349398700530689174854

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