Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} $
Sign $-0.999 - 0.0436i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (−1.5 − 0.866i)3-s + (1.73 − i)4-s + (−0.267 + i)5-s + (2.36 + 0.633i)6-s + (−2.36 − 1.36i)7-s + (−1.99 + 2i)8-s + (1.5 + 2.59i)9-s − 1.46i·10-s + (−4.23 + 1.13i)11-s − 3.46·12-s + (−3.36 − 0.901i)13-s + (3.73 + 0.999i)14-s + (1.26 − 1.26i)15-s + (1.99 − 3.46i)16-s − 5.73·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.866 − 0.499i)3-s + (0.866 − 0.5i)4-s + (−0.119 + 0.447i)5-s + (0.965 + 0.258i)6-s + (−0.894 − 0.516i)7-s + (−0.707 + 0.707i)8-s + (0.5 + 0.866i)9-s − 0.462i·10-s + (−1.27 + 0.341i)11-s − 0.999·12-s + (−0.933 − 0.250i)13-s + (0.997 + 0.267i)14-s + (0.327 − 0.327i)15-s + (0.499 − 0.866i)16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(144\)    =    \(2^{4} \cdot 3^{2}\)
\( \varepsilon \)  =  $-0.999 - 0.0436i$
motivic weight  =  \(1\)
character  :  $\chi_{144} (133, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 144,\ (\ :1/2),\ -0.999 - 0.0436i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
good5 \( 1 + (0.267 - i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.36 + 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.23 - 1.13i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.36 + 0.901i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 5.73T + 17T^{2} \)
19 \( 1 + (2.36 - 2.36i)T - 19iT^{2} \)
23 \( 1 + (-4.09 + 2.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.633 + 2.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (0.267 + 0.464i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.73 - 4.73i)T + 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (8.33 - 2.23i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.83 + 6.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.46 + 7.46i)T + 53iT^{2} \)
59 \( 1 + (1.96 - 7.33i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3 + 11.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.59 - 1.76i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 - 6.26iT - 73T^{2} \)
79 \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.366 - 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2iT - 89T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−12.57458115351867995140311781522, −11.23282193544176049839975743328, −10.51747393909866444324423786089, −9.766870970907271383751969584107, −8.139057893690085841848224931074, −7.08865245648435220077871456057, −6.50269972909689479506041912865, −5.03899878026064847581236989605, −2.50903389517737029009421037721, 0, 2.79700404624231203986260100087, 4.74503044809173584197382618871, 6.15713347037993703936757882100, 7.24309662101516330627775025940, 8.803182800278576681178790141765, 9.493053514051915962514721438516, 10.59849603715670775130457849865, 11.29987979499578249116801848688, 12.54018850008471894709126549214

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