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Properties

Label 2-2268-63.58-c1-0-8
Degree $2$
Conductor $2268$
Sign $0.999 + 0.0354i$
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  + (−2.00 − 3.47i)5-s + (−1.89 + 1.84i)7-s + (0.885 − 1.53i)11-s + (−0.114 + 0.198i)13-s + (3.04 + 5.27i)17-s + (−3.27 + 5.67i)19-s + (0.769 + 1.33i)23-s + (−5.55 + 9.62i)25-s + (0.271 + 0.469i)29-s + 4.55·31-s + (10.2 + 2.86i)35-s + (1.54 − 2.66i)37-s + (4.43 − 7.69i)41-s + (−2.12 − 3.67i)43-s − 0.757·47-s + ⋯
L(s)  = 1  + (−0.897 − 1.55i)5-s + (−0.715 + 0.698i)7-s + (0.267 − 0.462i)11-s + (−0.0317 + 0.0549i)13-s + (0.739 + 1.28i)17-s + (−0.752 + 1.30i)19-s + (0.160 + 0.277i)23-s + (−1.11 + 1.92i)25-s + (0.0503 + 0.0872i)29-s + 0.817·31-s + (1.72 + 0.484i)35-s + (0.253 − 0.438i)37-s + (0.693 − 1.20i)41-s + (−0.323 − 0.560i)43-s − 0.110·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.999 + 0.0354i$
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: \((2,\ 2268,\ (\ :1/2),\ 0.999 + 0.0354i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.173886624\)
\(L(\frac12)\) \(\approx\) \(1.173886624\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.89 - 1.84i)T \)
good5 \( 1 + (2.00 + 3.47i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.885 + 1.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.114 - 0.198i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.04 - 5.27i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.27 - 5.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.769 - 1.33i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.271 - 0.469i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 + (-1.54 + 2.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.43 + 7.69i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.757T + 47T^{2} \)
53 \( 1 + (3.19 + 5.53i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 5.17T + 59T^{2} \)
61 \( 1 - 12.5T + 61T^{2} \)
67 \( 1 + 6.18T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + (5.08 + 8.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (-8.66 - 15.0i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (5.04 - 8.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.91 - 8.50i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.851413996036036358783148144834, −8.307664858054532662622399289419, −7.83318694672567656772135386298, −6.55566554329002466176940494952, −5.76176291975602524304829360125, −5.14037039234412391266273016720, −3.91796135469996504475415347449, −3.63069689372264375410001919290, −2.00587021909282220706459695606, −0.794302985389181549296926861773, 0.60625718088102149626618487091, 2.62489801372902723628624452862, 3.09494794740767019956205277902, 4.07279140431633995159834112035, 4.81988652290861301554137282913, 6.34188616838061545163851131999, 6.72629448868744916537278443093, 7.38886901844042317770446777731, 7.960656910820860828972575958614, 9.144520690992196915682417034043

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