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Properties

Label 2-2268-63.47-c1-0-10
Degree $2$
Conductor $2268$
Sign $0.110 - 0.993i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + 3·5-s + (0.5 + 2.59i)7-s + 5.19i·11-s + (−1.5 − 2.59i)17-s + (1.5 + 0.866i)19-s + 5.19i·23-s + 4·25-s + (1.5 + 0.866i)31-s + (1.5 + 7.79i)35-s + (−3.5 + 6.06i)37-s + (−3 − 5.19i)41-s + (−2 + 3.46i)43-s + (−1.5 − 2.59i)47-s + (−6.5 + 2.59i)49-s + (−4.5 + 2.59i)53-s + ⋯
L(s)  = 1  + 1.34·5-s + (0.188 + 0.981i)7-s + 1.56i·11-s + (−0.363 − 0.630i)17-s + (0.344 + 0.198i)19-s + 1.08i·23-s + 0.800·25-s + (0.269 + 0.155i)31-s + (0.253 + 1.31i)35-s + (−0.575 + 0.996i)37-s + (−0.468 − 0.811i)41-s + (−0.304 + 0.528i)43-s + (−0.218 − 0.378i)47-s + (−0.928 + 0.371i)49-s + (−0.618 + 0.356i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.110 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.110 - 0.993i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.110 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.162030126\)
\(L(\frac12)\) \(\approx\) \(2.162030126\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 - 5.19iT - 11T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 2.59i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.5 + 6.06i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (10.5 - 6.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.5 + 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.364789242020456460814057884160, −8.621309308469259514613133221275, −7.58524307345143014828026318324, −6.80562475031063690875275129489, −6.02066750183582009935559786758, −5.20773698191791038252651568163, −4.72618983539529281305414390028, −3.24130408263612626306112598333, −2.17600248229156634412609471073, −1.65958147445020219223360497828, 0.72023189104930282554618849696, 1.84229964395840046097251360452, 2.95778013428446797122959932101, 3.92082740073661750972298074271, 4.94583455079329553210834293689, 5.83090593779785982239475053500, 6.37316710686069521372408838946, 7.17672664451440366856367127570, 8.281961484380062495765253375445, 8.752041133443531888829114951903

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