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Properties

Label 2-12e2-144.133-c1-0-1
Degree $2$
Conductor $144$
Sign $-0.331 - 0.943i$
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  + (0.327 + 1.37i)2-s + (−0.844 − 1.51i)3-s + (−1.78 + 0.900i)4-s + (−0.891 + 3.32i)5-s + (1.80 − 1.65i)6-s + (3.95 + 2.28i)7-s + (−1.82 − 2.16i)8-s + (−1.57 + 2.55i)9-s + (−4.86 − 0.138i)10-s + (−2.12 + 0.568i)11-s + (2.86 + 1.93i)12-s + (−0.0649 − 0.0174i)13-s + (−1.84 + 6.19i)14-s + (5.78 − 1.46i)15-s + (2.37 − 3.21i)16-s − 0.00952·17-s + ⋯
L(s)  = 1  + (0.231 + 0.972i)2-s + (−0.487 − 0.872i)3-s + (−0.892 + 0.450i)4-s + (−0.398 + 1.48i)5-s + (0.736 − 0.676i)6-s + (1.49 + 0.863i)7-s + (−0.644 − 0.764i)8-s + (−0.524 + 0.851i)9-s + (−1.53 − 0.0436i)10-s + (−0.639 + 0.171i)11-s + (0.828 + 0.560i)12-s + (−0.0180 − 0.00482i)13-s + (−0.494 + 1.65i)14-s + (1.49 − 0.377i)15-s + (0.594 − 0.803i)16-s − 0.00231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.331 - 0.943i$
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: \((2,\ 144,\ (\ :1/2),\ -0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559272 + 0.789690i\)
\(L(\frac12)\) \(\approx\) \(0.559272 + 0.789690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 - 1.37i)T \)
3 \( 1 + (0.844 + 1.51i)T \)
good5 \( 1 + (0.891 - 3.32i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-3.95 - 2.28i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.12 - 0.568i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.0649 + 0.0174i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 0.00952T + 17T^{2} \)
19 \( 1 + (-2.79 + 2.79i)T - 19iT^{2} \)
23 \( 1 + (-4.81 + 2.78i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.343 - 1.28i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.30 + 2.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.32 + 3.32i)T + 37iT^{2} \)
41 \( 1 + (-4.34 + 2.50i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.68 + 0.718i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.77 - 4.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.80 - 1.80i)T + 53iT^{2} \)
59 \( 1 + (0.0592 - 0.221i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.91 - 10.8i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.45 + 1.19i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.21iT - 71T^{2} \)
73 \( 1 - 6.99iT - 73T^{2} \)
79 \( 1 + (-6.86 + 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.18 + 11.8i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.28iT - 89T^{2} \)
97 \( 1 + (4.69 - 8.12i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.64525243455214085019618057383, −12.43064313506600407522059297528, −11.46901051987054610990211131570, −10.70584942880261165704515395995, −8.781781359317048989946230153179, −7.67209721273952001643460406559, −7.16102102881686275791083270737, −5.89179396887739854070981951244, −4.87095813468885882874418321851, −2.68695137991744985013307939998, 1.10197584782988246405768567724, 3.79792940350450381479372950457, 4.86048600514155481089032030927, 5.29047315475895443739875398030, 7.944817931169800214724034376334, 8.819714161460952751145479237650, 9.943425387695028702486775284732, 10.98515403907098885672131884114, 11.60930814740281566171771191075, 12.55198076466425639386292441224

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