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Properties

Label 2-2268-63.4-c1-0-23
Degree $2$
Conductor $2268$
Sign $-0.0788 + 0.996i$
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  + 2·5-s + (−2.5 + 0.866i)7-s − 2·11-s + (1.5 − 2.59i)13-s + (4 − 6.92i)17-s + (0.5 + 0.866i)19-s − 8·23-s − 25-s + (2 + 3.46i)29-s + (−1.5 − 2.59i)31-s + (−5 + 1.73i)35-s + (0.5 + 0.866i)37-s + (3 − 5.19i)41-s + (−5.5 − 9.52i)43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + 0.894·5-s + (−0.944 + 0.327i)7-s − 0.603·11-s + (0.416 − 0.720i)13-s + (0.970 − 1.68i)17-s + (0.114 + 0.198i)19-s − 1.66·23-s − 0.200·25-s + (0.371 + 0.643i)29-s + (−0.269 − 0.466i)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 + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.0788 + 0.996i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.0788 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.297769049\)
\(L(\frac12)\) \(\approx\) \(1.297769049\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 - 2T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
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 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (-2 - 3.46i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \)
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 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
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

−9.000212758456728362621690504352, −7.996509242237018635867543329055, −7.31060512349369400305750995214, −6.30549875346230478326269624513, −5.66533765327689565967496171376, −5.14810240008522480478616670969, −3.71632197440089622758523613608, −2.91737663209407363230463544992, −2.02380813467065873040384379180, −0.43432404837003469094928763357, 1.38492303028493436244175965900, 2.39962699984127349239247339258, 3.52510699373237300541551136865, 4.25811942113176240075701198402, 5.53513632282713089785832547663, 6.14493559053094399184237769290, 6.62503749218396645330216282857, 7.84179929845912335778906022086, 8.344517441363426941166211926395, 9.515301336661274854884769863754

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