Properties

Label 2-2736-57.56-c1-0-15
Degree $2$
Conductor $2736$
Sign $0.662 - 0.749i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + 1.41i·5-s − 2·7-s − 1.41i·11-s + 1.41i·17-s + (−1 − 4.24i)19-s − 1.41i·23-s + 2.99·25-s + 6·29-s − 2.82i·35-s + 8.48i·37-s + 6·41-s + 4·43-s + 7.07i·47-s − 3·49-s + 6·53-s + ⋯
L(s)  = 1  + 0.632i·5-s − 0.755·7-s − 0.426i·11-s + 0.342i·17-s + (−0.229 − 0.973i)19-s − 0.294i·23-s + 0.599·25-s + 1.11·29-s − 0.478i·35-s + 1.39i·37-s + 0.937·41-s + 0.609·43-s + 1.03i·47-s − 0.428·49-s + 0.824·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.662 - 0.749i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.495850724\)
\(L(\frac12)\) \(\approx\) \(1.495850724\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (1 + 4.24i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 8.48iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 7.07iT - 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 4T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 - 8.48iT - 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 8.48iT - 97T^{2} \)
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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

−8.900840881034435884539232854566, −8.247565901398626653927898346463, −7.27280471617382105116998514886, −6.56491759360658942569256961787, −6.11957575570286189759501682185, −5.00652095774585820789895156975, −4.12270903404328261878565353584, −3.06769429339843665983831725370, −2.56595461323032056866578078774, −0.932692176303675529087860756415, 0.62343445286279761208255850052, 1.92858717990471406488940216022, 3.03465511737228209844666263977, 3.97679949688989378860562759644, 4.78561051033238522822154556347, 5.65670085924131563834963298536, 6.39907636062997152988100109556, 7.23262651496418731159759532417, 7.968607039488244870944908177974, 8.862011073659789189859265964456

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