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Properties

Label 2-726-11.9-c1-0-11
Degree $2$
Conductor $726$
Sign $0.882 + 0.469i$
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 + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (0.618 + 1.90i)5-s + (0.309 + 0.951i)6-s + (−3.23 − 2.35i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s − 1.99·10-s − 0.999·12-s + (1.85 − 5.70i)13-s + (3.23 − 2.35i)14-s + (1.61 + 1.17i)15-s + (0.309 + 0.951i)16-s + (−0.618 − 1.90i)17-s + (0.809 + 0.587i)18-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 − 0.293i)4-s + (0.276 + 0.850i)5-s + (0.126 + 0.388i)6-s + (−1.22 − 0.888i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s − 0.632·10-s − 0.288·12-s + (0.514 − 1.58i)13-s + (0.864 − 0.628i)14-s + (0.417 + 0.303i)15-s + (0.0772 + 0.237i)16-s + (−0.149 − 0.461i)17-s + (0.190 + 0.138i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.882 + 0.469i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28367 - 0.320438i\)
\(L(\frac12)\) \(\approx\) \(1.28367 - 0.320438i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good5 \( 1 + (-0.618 - 1.90i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (3.23 + 2.35i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.85 + 5.70i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.618 + 1.90i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-4.85 - 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (4.85 + 3.52i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (4.85 - 3.52i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-9.70 + 7.05i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.618 + 1.90i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.70 + 7.05i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.32 - 13.3i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (3.70 + 11.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.85 + 3.52i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.23 + 3.80i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.23 + 3.80i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (4.32 - 13.3i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.38742319702607374521492557037, −9.423434779818864423712252423412, −8.560772471842898255862659753818, −7.45357915411923691371971266190, −6.94509810708132291906613806686, −6.24882969897932782515026880545, −5.10302874296492397685506445436, −3.48358029575819357009308847719, −2.91235742062004568964894088485, −0.74973266678529061496343877877, 1.52365636070987427475540665166, 2.78390143034955716429162935229, 3.79050882073325751712452197254, 4.83757683351575665102165089222, 5.93790377519816496494005664003, 6.95652619955459433972446286041, 8.434331800088014902295768890693, 8.959988472564092582342182719379, 9.478773466264669439795240497142, 10.20289555200772000805147224542

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