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Properties

Label 2-12e2-144.13-c1-0-13
Degree $2$
Conductor $144$
Sign $-0.999 + 0.0436i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.999 + 0.0436i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.999 + 0.0436i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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