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Properties

Label 2-12e2-144.131-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.411 + 0.911i$
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  + (−0.0415 − 1.41i)2-s + (1.52 − 0.819i)3-s + (−1.99 + 0.117i)4-s + (−0.769 + 0.206i)5-s + (−1.22 − 2.12i)6-s + (−2.17 − 3.76i)7-s + (0.248 + 2.81i)8-s + (1.65 − 2.50i)9-s + (0.323 + 1.07i)10-s + (3.93 + 1.05i)11-s + (−2.94 + 1.81i)12-s + (1.69 − 0.454i)13-s + (−5.23 + 3.23i)14-s + (−1.00 + 0.945i)15-s + (3.97 − 0.468i)16-s + 6.68i·17-s + ⋯
L(s)  = 1  + (−0.0293 − 0.999i)2-s + (0.880 − 0.473i)3-s + (−0.998 + 0.0586i)4-s + (−0.344 + 0.0922i)5-s + (−0.498 − 0.866i)6-s + (−0.822 − 1.42i)7-s + (0.0879 + 0.996i)8-s + (0.551 − 0.833i)9-s + (0.102 + 0.341i)10-s + (1.18 + 0.318i)11-s + (−0.851 + 0.524i)12-s + (0.470 − 0.126i)13-s + (−1.39 + 0.863i)14-s + (−0.259 + 0.244i)15-s + (0.993 − 0.117i)16-s + 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.411 + 0.911i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650437 - 1.00676i\)
\(L(\frac12)\) \(\approx\) \(0.650437 - 1.00676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0415 + 1.41i)T \)
3 \( 1 + (-1.52 + 0.819i)T \)
good5 \( 1 + (0.769 - 0.206i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3.93 - 1.05i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.69 + 0.454i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 6.68iT - 17T^{2} \)
19 \( 1 + (0.708 - 0.708i)T - 19iT^{2} \)
23 \( 1 + (-3.88 - 2.24i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.98 + 1.06i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-4.94 - 2.85i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.51 - 1.51i)T - 37iT^{2} \)
41 \( 1 + (1.36 - 2.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.30 + 8.60i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.23 + 2.13i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.68 + 1.68i)T + 53iT^{2} \)
59 \( 1 + (0.269 + 1.00i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.528 - 1.97i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.14 + 8.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 8.05iT - 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + (11.9 - 6.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.817 + 3.05i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.71T + 89T^{2} \)
97 \( 1 + (-3.66 - 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.93063430074625376207651677561, −11.89571317659809961700860930574, −10.65324562508837732919025920922, −9.784313355087415086829148555041, −8.783910907681865957766117027383, −7.64307641808238286493046595871, −6.48185486926100985752524123329, −3.98456394019318337400874679321, −3.54224284925202065604526024987, −1.42121077885755049292857623763, 3.07381775016345092638191665480, 4.44908162196483485245002625365, 5.86959748232894660539621797494, 7.03839842748786765573771702216, 8.408755229183968873409021304675, 9.136817401962261611052161574395, 9.671250187590586504877722307262, 11.52569359035748652147354427185, 12.69794836079509008002595537327, 13.71329372572036766098142473855

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