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Properties

Label 2-12e2-144.133-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 + 0.366i)2-s + (−0.866 + 1.5i)3-s + (1.73 − i)4-s + (1 − 3.73i)5-s + (0.633 − 2.36i)6-s + (0.633 + 0.366i)7-s + (−1.99 + 2i)8-s + (−1.5 − 2.59i)9-s + 5.46i·10-s + (2.86 − 0.767i)11-s + 3.46i·12-s + (6.09 + 1.63i)13-s + (−1 − 0.267i)14-s + (4.73 + 4.73i)15-s + (1.99 − 3.46i)16-s − 2.26·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (−0.499 + 0.866i)3-s + (0.866 − 0.5i)4-s + (0.447 − 1.66i)5-s + (0.258 − 0.965i)6-s + (0.239 + 0.138i)7-s + (−0.707 + 0.707i)8-s + (−0.5 − 0.866i)9-s + 1.72i·10-s + (0.864 − 0.231i)11-s + 0.999i·12-s + (1.69 + 0.453i)13-s + (−0.267 − 0.0716i)14-s + (1.22 + 1.22i)15-s + (0.499 − 0.866i)16-s − 0.550·17-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: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.999 + 0.0436i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.723787 - 0.0157930i\)
\(L(\frac12)\) \(\approx\) \(0.723787 - 0.0157930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T \)
good5 \( 1 + (-1 + 3.73i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.633 - 0.366i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.86 + 0.767i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-6.09 - 1.63i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 2.26T + 17T^{2} \)
19 \( 1 + (0.633 - 0.633i)T - 19iT^{2} \)
23 \( 1 + (-1.09 + 0.633i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.633 - 2.36i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.73 + 6.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 - 1.26i)T + 37iT^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.23 - 0.330i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.83 - 8.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.535 + 0.535i)T + 53iT^{2} \)
59 \( 1 + (1.33 - 4.96i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.803 - 3i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.23 + 1.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.9iT - 71T^{2} \)
73 \( 1 - 9.73iT - 73T^{2} \)
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 \( 1 + (4.13 - 7.16i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.94690318904039083142615001477, −11.68774020176796013091594321489, −11.05200634195610041229649991239, −9.690240391310992884490074308370, −8.940123209963376357906373123593, −8.451266136949706590916681764442, −6.37797699913261667604710731909, −5.54086619136761879072610801135, −4.17803090641290544437846190947, −1.27186660900970486748029768107, 1.74576706969998869124035091338, 3.28754684796516277935594559354, 6.15734370009165966506239904039, 6.67459875878768755936555091578, 7.68547276714428948630492265150, 8.925814671999008289510159098312, 10.40045798648931944374885465453, 11.00817660880283013059592830398, 11.64503276358115957668478127624, 13.02144481085205742291811681236

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