Properties

Degree $2$
Conductor $2268$
Sign $0.952 - 0.305i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 1.5i)5-s + (2.62 − 0.358i)7-s + (−3.67 + 2.12i)11-s + (2.12 + 1.22i)13-s − 1.73·17-s + 2.44i·19-s + (7.34 + 4.24i)23-s + (1 + 1.73i)25-s + (−3.67 + 2.12i)29-s + (6.36 + 3.67i)31-s + (1.73 − 4.24i)35-s + 37-s + (−0.866 + 1.5i)41-s + (−3.5 − 6.06i)43-s + (6.06 + 10.5i)47-s + ⋯
L(s)  = 1  + (0.387 − 0.670i)5-s + (0.990 − 0.135i)7-s + (−1.10 + 0.639i)11-s + (0.588 + 0.339i)13-s − 0.420·17-s + 0.561i·19-s + (1.53 + 0.884i)23-s + (0.200 + 0.346i)25-s + (−0.682 + 0.393i)29-s + (1.14 + 0.659i)31-s + (0.292 − 0.717i)35-s + 0.164·37-s + (−0.135 + 0.234i)41-s + (−0.533 − 0.924i)43-s + (0.884 + 1.53i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.952 - 0.305i$
Motivic weight: \(1\)
Character: $\chi_{2268} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.093321383\)
\(L(\frac12)\) \(\approx\) \(2.093321383\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.62 + 0.358i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.67 - 2.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.12 - 1.22i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.73T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 + (-7.34 - 4.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.67 - 2.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.36 - 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (0.866 - 1.5i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.5 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.06 - 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-4.33 + 7.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.12 + 1.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.06 + 10.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (2.12 - 1.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.984893543271479215569862300951, −8.346742892031484553681414517795, −7.57826227221924695210614458269, −6.86607501071026925700754265455, −5.67398716419400583616996177967, −5.05960415593083677990536019658, −4.48185857612643045688839476607, −3.25199121856420309671050234511, −2.01260358895977304152859967400, −1.18291154028982388960686095866, 0.833529948232097098051280884403, 2.37063616686154375997038311165, 2.87823328155389908151593711960, 4.20115194507462246049905576623, 5.10686019689295075181035015551, 5.77722530990046798383585980366, 6.68055981004230422503274085548, 7.42777558203863289727155070233, 8.441252155463257811158542277791, 8.638624161595955220778431353484

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