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Properties

Label 2-726-11.5-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 + (−1.23 + 3.80i)5-s + (−0.309 + 0.951i)6-s + (−1.61 + 1.17i)7-s + (0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + 3.99·10-s + 0.999·12-s + (−1.23 − 3.80i)13-s + (1.61 + 1.17i)14-s + (3.23 − 2.35i)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 + (−0.552 + 1.70i)5-s + (−0.126 + 0.388i)6-s + (−0.611 + 0.444i)7-s + (0.286 + 0.207i)8-s + (0.103 + 0.317i)9-s + 1.26·10-s + 0.288·12-s + (−0.342 − 1.05i)13-s + (0.432 + 0.314i)14-s + (0.835 − 0.607i)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: $\chi_{726} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 726,\ (\ :1/2),\ -0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0616631 - 0.247023i\)
\(L(\frac12)\) \(\approx\) \(0.0616631 - 0.247023i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 - 3.80i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.61 - 1.17i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.23 + 3.80i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.618 + 1.90i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-8.09 + 5.87i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.47 + 7.60i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.61 + 1.17i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.61 - 1.17i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-1.61 - 1.17i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.23 - 3.80i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.47 + 7.60i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + (-0.618 + 1.90i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.85 - 3.52i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.09 + 9.51i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.23 - 3.80i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (0.618 + 1.90i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.21434272507630648732621899587, −9.583911997740043228371283495252, −8.055485772037303255871592461319, −7.56162304763252386938252625452, −6.48136389154395999889901316683, −5.78834186648881317117218020333, −4.22592083276388511284518728061, −3.07079915875883433441105871574, −2.43046996038888647250384531366, −0.16136384653226340996571649667, 1.31343638831302220106744895570, 3.77584003588981457796164081637, 4.55009859342003340383578753982, 5.27835766394675771721676027711, 6.37017077622831097669911532029, 7.25277901514010945405430483662, 8.366906192212396060283891311057, 8.896132272781126716509753540508, 9.763218065252350518344863447689, 10.49991941307316745392450473643

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