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Properties

Label 2-726-11.4-c1-0-10
Degree $2$
Conductor $726$
Sign $0.530 + 0.847i$
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 + (0.809 − 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s − 12-s + (−4.85 − 3.52i)13-s + (−0.809 + 0.587i)16-s + (4.85 − 3.52i)17-s + (0.309 + 0.951i)18-s + (1.85 − 5.70i)19-s + 6·23-s + (0.809 + 0.587i)24-s + (−1.54 + 4.75i)25-s + (1.85 + 5.70i)26-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.330 − 0.239i)6-s + (0.109 − 0.336i)8-s + (−0.269 − 0.195i)9-s − 0.288·12-s + (−1.34 − 0.978i)13-s + (−0.202 + 0.146i)16-s + (1.17 − 0.855i)17-s + (0.0728 + 0.224i)18-s + (0.425 − 1.30i)19-s + 1.25·23-s + (0.165 + 0.119i)24-s + (−0.309 + 0.951i)25-s + (0.363 + 1.11i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(726\)    =    \(2 \cdot 3 \cdot 11^{2}\)
Sign: $0.530 + 0.847i$
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: \((2,\ 726,\ (\ :1/2),\ 0.530 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.811706 - 0.449787i\)
\(L(\frac12)\) \(\approx\) \(0.811706 - 0.449787i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.85 + 3.52i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.85 + 3.52i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.85 + 5.70i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.23 + 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.618 + 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.85 + 5.70i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.70 - 7.05i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.70 + 11.4i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.85 - 3.52i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + (-4.85 + 3.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.70 + 11.4i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.70 + 7.05i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-8.09 - 5.87i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.28754807334576343804349448515, −9.379939171466437942410073782622, −8.949803625671351266705920804326, −7.48681324372664095851158235385, −7.21470240606940323090862669380, −5.48318922982578825324048252486, −4.93452786249802920145580765404, −3.43075621527987525255836391950, −2.61411192013216134478651136062, −0.67127629307202107240389362033, 1.27380738764506849492894004922, 2.58488732123375591164228831450, 4.20771125128313108507751941814, 5.43727253820577733392520736357, 6.22181870961698285942269078337, 7.23249475353295942279467358422, 7.79279733559829037980161029844, 8.708268335824072860533124381955, 9.747168378507475502308117010753, 10.26586510151714165025822704890

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