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Properties

Label 2-726-11.4-c1-0-3
Degree $2$
Conductor $726$
Sign $0.751 - 0.659i$
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  + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (−1.61 + 1.17i)5-s + (0.809 − 0.587i)6-s + (−1.23 − 3.80i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + 2·10-s − 12-s + (4.85 + 3.52i)13-s + (−1.23 + 3.80i)14-s + (−0.618 − 1.90i)15-s + (−0.809 + 0.587i)16-s + (−1.61 + 1.17i)17-s + (0.309 + 0.951i)18-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (−0.723 + 0.525i)5-s + (0.330 − 0.239i)6-s + (−0.467 − 1.43i)7-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s + 0.632·10-s − 0.288·12-s + (1.34 + 0.978i)13-s + (−0.330 + 1.01i)14-s + (−0.159 − 0.491i)15-s + (−0.202 + 0.146i)16-s + (−0.392 + 0.285i)17-s + (0.0728 + 0.224i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(5.79713\)
Root analytic conductor: \(2.40772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{726} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.768908 + 0.289348i\)
\(L(\frac12)\) \(\approx\) \(0.768908 + 0.289348i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (1.61 - 1.17i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.23 + 3.80i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-4.85 - 3.52i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.61 - 1.17i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.23 + 3.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.85 - 5.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.85 - 5.70i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (3.70 - 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.61 + 1.17i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.70 - 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-11.3 + 8.22i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-9.70 + 7.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.85 + 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-3.23 - 2.35i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (3.23 - 2.35i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (-11.3 - 8.22i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.78943182047365696812061155813, −9.700841675042308616280764568070, −8.971473155130840006567766323548, −7.976494597289888650958210934837, −6.95902023664703692712328894277, −6.49769950797006363371599891045, −4.67534136306998914680482195795, −3.82138823649426014145149080488, −3.13491815482571287481930262944, −1.08857327621359780693478627243, 0.66865795513221408564615882480, 2.32191477131795182975973936129, 3.69143159548106604838069279313, 5.29893744985161883649625955473, 5.88733775518371041778884946794, 6.79330018614270107057396536910, 7.990681282448467138872239926946, 8.442922514200025011009876155194, 9.132936745606675923757315170459, 10.20021772018958682208889836867

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