Properties

Label 2-2268-63.58-c1-0-19
Degree $2$
Conductor $2268$
Sign $0.415 + 0.909i$
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 + (2 − 1.73i)7-s + (1 − 1.73i)11-s + (1.5 − 2.59i)13-s + (4 + 6.92i)17-s + (0.5 − 0.866i)19-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + (2 + 3.46i)29-s + 3·31-s + (−5 − 1.73i)35-s + (0.5 − 0.866i)37-s + (3 − 5.19i)41-s + (−5.5 − 9.52i)43-s − 6·47-s + ⋯
L(s)  = 1  + (−0.447 − 0.774i)5-s + (0.755 − 0.654i)7-s + (0.301 − 0.522i)11-s + (0.416 − 0.720i)13-s + (0.970 + 1.68i)17-s + (0.114 − 0.198i)19-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + (0.371 + 0.643i)29-s + 0.538·31-s + (−0.845 − 0.292i)35-s + (0.0821 − 0.142i)37-s + (0.468 − 0.811i)41-s + (−0.838 − 1.45i)43-s − 0.875·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.415 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.415 + 0.909i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.415 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.946653573\)
\(L(\frac12)\) \(\approx\) \(1.946653573\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4 - 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + (-1 - 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.484636550045670076941386799302, −8.331914229154917331808340038655, −7.53246245241911807888563446648, −6.58006057598810795509374823461, −5.52621592511868185873439538250, −4.98893842650271006639456741843, −3.86709123919351988825222384290, −3.38796596737608600423443744943, −1.61824649618290751723304394337, −0.826001396944456149174647900534, 1.20498453161441346019652576867, 2.53850821728945269921201718714, 3.21140986362217815392390639253, 4.53217634924388527413796181093, 4.96912370270836326265886419669, 6.22948151303062848260184101696, 6.80859673526925755125187882360, 7.69023672038116709112585594124, 8.242043452222187806673227761602, 9.285702117274318826432183164128

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