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Properties

Label 2-12e2-144.11-c1-0-1
Degree $2$
Conductor $144$
Sign $-0.764 + 0.644i$
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  + (0.0484 + 1.41i)2-s + (−1.65 + 0.517i)3-s + (−1.99 + 0.136i)4-s + (−1.94 − 0.521i)5-s + (−0.811 − 2.31i)6-s + (−0.322 + 0.558i)7-s + (−0.290 − 2.81i)8-s + (2.46 − 1.71i)9-s + (0.642 − 2.77i)10-s + (−5.59 + 1.49i)11-s + (3.22 − 1.25i)12-s + (−1.97 − 0.530i)13-s + (−0.804 − 0.428i)14-s + (3.48 − 0.144i)15-s + (3.96 − 0.546i)16-s + 3.05i·17-s + ⋯
L(s)  = 1  + (0.0342 + 0.999i)2-s + (−0.954 + 0.298i)3-s + (−0.997 + 0.0684i)4-s + (−0.869 − 0.233i)5-s + (−0.331 − 0.943i)6-s + (−0.121 + 0.210i)7-s + (−0.102 − 0.994i)8-s + (0.821 − 0.570i)9-s + (0.203 − 0.877i)10-s + (−1.68 + 0.451i)11-s + (0.931 − 0.363i)12-s + (−0.548 − 0.147i)13-s + (−0.214 − 0.114i)14-s + (0.899 − 0.0373i)15-s + (0.990 − 0.136i)16-s + 0.741i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.764 + 0.644i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.764 + 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0740498 - 0.202866i\)
\(L(\frac12)\) \(\approx\) \(0.0740498 - 0.202866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0484 - 1.41i)T \)
3 \( 1 + (1.65 - 0.517i)T \)
good5 \( 1 + (1.94 + 0.521i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.322 - 0.558i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.59 - 1.49i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.97 + 0.530i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.05iT - 17T^{2} \)
19 \( 1 + (-4.11 - 4.11i)T + 19iT^{2} \)
23 \( 1 + (7.05 - 4.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.49 + 1.20i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-0.147 + 0.0854i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.65 + 2.65i)T + 37iT^{2} \)
41 \( 1 + (-0.983 - 1.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.220 + 0.821i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-4.02 + 6.97i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.25 - 3.25i)T - 53iT^{2} \)
59 \( 1 + (-0.135 + 0.506i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.805 + 3.00i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.583 + 2.17i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 - 9.33iT - 73T^{2} \)
79 \( 1 + (9.44 + 5.45i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.31 - 12.3i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.86T + 89T^{2} \)
97 \( 1 + (7.33 - 12.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.75180358320000947849145657257, −12.51187199065315128879884005379, −12.05844425099175605038028311072, −10.45473920766927120150286170501, −9.726467238857721511306497723418, −8.062997298080243668651533985607, −7.48039549440603586211743857622, −5.94833551484452202946850914343, −5.09499637957299016769510121562, −3.90636923596638983549079528261, 0.23353789861414415743521203947, 2.76355475893234254263647615034, 4.47200822238369028711745576840, 5.48770284783378825390478254308, 7.24941391008693426272940454260, 8.187246567908390507474572360360, 9.885203089960173508974113982471, 10.66805109168007543915362667680, 11.54428422980098866050612172920, 12.20697931399986255751030497345

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