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Properties

Label 2-2268-7.2-c1-0-17
Degree $2$
Conductor $2268$
Sign $0.991 + 0.126i$
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  + (1 − 1.73i)5-s + (0.5 − 2.59i)7-s + (2 + 3.46i)11-s + 3·13-s + (3.5 + 6.06i)17-s + (−2.5 + 4.33i)19-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 29-s + (1.5 + 2.59i)31-s + (−4 − 3.46i)35-s + (−5.5 + 9.52i)37-s + 9·41-s + 5·43-s + (1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.188 − 0.981i)7-s + (0.603 + 1.04i)11-s + 0.832·13-s + (0.848 + 1.47i)17-s + (−0.573 + 0.993i)19-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.185·29-s + (0.269 + 0.466i)31-s + (−0.676 − 0.585i)35-s + (−0.904 + 1.56i)37-s + 1.40·41-s + 0.762·43-s + (0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.238864694\)
\(L(\frac12)\) \(\approx\) \(2.238864694\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.5 - 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 7.79i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + T + 83T^{2} \)
89 \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−8.942208119518173810111571822212, −8.280659072023465144555160046524, −7.56356873058459653499554344372, −6.56096757619811273597365574954, −5.98584445176522998021852905155, −4.87643816792581559154838464682, −4.21065969920295377085691067486, −3.43221138078520902183173176862, −1.73681416521329929421670475108, −1.18744018028661562197458210956, 0.955309841623043748884814650501, 2.44864328532749688815154596024, 3.01073449433670574788753604885, 4.08438374062294915766253678737, 5.35826424940150140924537223401, 5.85388411641229230669263906141, 6.63251542975325556312622528975, 7.41305851032987091994611554253, 8.413451260524724988674982134002, 9.131964889941654984703736198573

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