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Properties

Label 2-726-33.2-c1-0-32
Degree $2$
Conductor $726$
Sign $-0.773 + 0.634i$
Analytic cond. $5.79713$
Root an. cond. $2.40772$
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  + (−0.309 + 0.951i)2-s + (−0.0222 − 1.73i)3-s + (−0.809 − 0.587i)4-s + (−1.34 + 0.437i)5-s + (1.65 + 0.514i)6-s + (2.49 − 3.43i)7-s + (0.809 − 0.587i)8-s + (−2.99 + 0.0770i)9-s − 1.41i·10-s + (−0.999 + 1.41i)12-s + (−4.03 − 1.31i)13-s + (2.49 + 3.43i)14-s + (0.786 + 2.31i)15-s + (0.309 + 0.951i)16-s + (0.853 − 2.87i)18-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (−0.0128 − 0.999i)3-s + (−0.404 − 0.293i)4-s + (−0.601 + 0.195i)5-s + (0.675 + 0.209i)6-s + (0.942 − 1.29i)7-s + (0.286 − 0.207i)8-s + (−0.999 + 0.0256i)9-s − 0.447i·10-s + (−0.288 + 0.408i)12-s + (−1.11 − 0.363i)13-s + (0.666 + 0.917i)14-s + (0.203 + 0.598i)15-s + (0.0772 + 0.237i)16-s + (0.201 − 0.677i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.773 + 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $-0.773 + 0.634i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ -0.773 + 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.207935 - 0.581422i\)
\(L(\frac12)\) \(\approx\) \(0.207935 - 0.581422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (0.0222 + 1.73i)T \)
11 \( 1 \)
good5 \( 1 + (1.34 - 0.437i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.49 + 3.43i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (4.03 + 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.23 - 3.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.61 + 1.17i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.85 + 3.52i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 + (5.81 + 8.00i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (6.72 + 2.18i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.65 - 9.15i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.03 - 1.31i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (-6.72 + 2.18i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.03 - 1.31i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.70 + 11.4i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + (-2.47 + 7.60i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.10470652910972827826097483953, −8.905893555474322073270635726866, −7.86941095875441595555082037363, −7.49414513945439718469054402632, −6.98625908384273942763968602905, −5.69824883317413835399893031047, −4.73385862540080734647169281256, −3.56352516336243622037379900636, −1.82602018325612161007012466836, −0.33445224909518010208599312905, 2.06488182160869593893861182237, 3.12293629647016340725159040479, 4.44820920383551541197257982718, 4.93237301545176002052286555046, 6.01325521334990510027706860418, 7.74508828281917857449586208229, 8.312324402837175162557978484706, 9.298965521195294229677099022485, 9.660619946328305521510178016642, 10.97731236504286375839883662853

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