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Properties

Label 2-2268-63.5-c1-0-4
Degree $2$
Conductor $2268$
Sign $-0.235 - 0.971i$
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  + (0.5 + 2.59i)7-s + (−6 − 3.46i)13-s + (7.5 + 4.33i)19-s + (2.5 − 4.33i)25-s + 8.66i·31-s + (−5 + 8.66i)37-s + (2.5 + 4.33i)43-s + (−6.5 + 2.59i)49-s + 15.5i·61-s − 16·67-s + (1.5 − 0.866i)73-s − 4·79-s + (6 − 17.3i)91-s + (−4.5 + 2.59i)97-s + (3 − 1.73i)103-s + ⋯
L(s)  = 1  + (0.188 + 0.981i)7-s + (−1.66 − 0.960i)13-s + (1.72 + 0.993i)19-s + (0.5 − 0.866i)25-s + 1.55i·31-s + (−0.821 + 1.42i)37-s + (0.381 + 0.660i)43-s + (−0.928 + 0.371i)49-s + 1.99i·61-s − 1.95·67-s + (0.175 − 0.101i)73-s − 0.450·79-s + (0.628 − 1.81i)91-s + (−0.456 + 0.263i)97-s + (0.295 − 0.170i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.235 - 0.971i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.235 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.245422092\)
\(L(\frac12)\) \(\approx\) \(1.245422092\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6 + 3.46i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-7.5 - 4.33i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + (5 - 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.5 - 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.5iT - 61T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.241120287501287822806023015155, −8.439838579284373965247057447371, −7.73571648716038545433327996182, −7.04772123572789125224173389924, −5.96191276069402467830350832969, −5.24922253637399933632372159944, −4.70436344929179661457289002605, −3.19903820334910438556399175602, −2.67547884331533200349383768652, −1.34222614519230771197340706868, 0.43707894927033003657397803710, 1.82958684189881850166730959991, 2.93674185631062289121744666526, 3.98894818898648604210386403960, 4.80374712206530339487832728401, 5.45534360400761364568114377388, 6.73974714735090711796008000802, 7.36897246798339752755832691041, 7.66553482348871477061744064262, 9.065719120598480542949080209965

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