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Properties

Label 2-12e2-9.5-c2-0-10
Degree $2$
Conductor $144$
Sign $-0.811 + 0.584i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.668 − 2.92i)3-s + (−0.0440 − 0.0254i)5-s + (−4.52 − 7.84i)7-s + (−8.10 + 3.91i)9-s + (−3.29 + 1.90i)11-s + (0.216 − 0.375i)13-s + (−0.0448 + 0.145i)15-s − 26.2i·17-s − 34.2·19-s + (−19.9 + 18.4i)21-s + (29.9 + 17.3i)23-s + (−12.4 − 21.6i)25-s + (16.8 + 21.0i)27-s + (14.0 − 8.10i)29-s + (17.1 − 29.7i)31-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)3-s + (−0.00880 − 0.00508i)5-s + (−0.647 − 1.12i)7-s + (−0.900 + 0.434i)9-s + (−0.299 + 0.172i)11-s + (0.0166 − 0.0288i)13-s + (−0.00299 + 0.00971i)15-s − 1.54i·17-s − 1.80·19-s + (−0.948 + 0.880i)21-s + (1.30 + 0.752i)23-s + (−0.499 − 0.865i)25-s + (0.624 + 0.781i)27-s + (0.483 − 0.279i)29-s + (0.553 − 0.959i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.811 + 0.584i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.270605 - 0.838709i\)
\(L(\frac12)\) \(\approx\) \(0.270605 - 0.838709i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.668 + 2.92i)T \)
good5 \( 1 + (0.0440 + 0.0254i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (4.52 + 7.84i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (3.29 - 1.90i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.216 + 0.375i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 26.2iT - 289T^{2} \)
19 \( 1 + 34.2T + 361T^{2} \)
23 \( 1 + (-29.9 - 17.3i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-14.0 + 8.10i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-17.1 + 29.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 29.2T + 1.36e3T^{2} \)
41 \( 1 + (-48.7 - 28.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-3.94 - 6.83i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-33.4 + 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 50.5iT - 2.80e3T^{2} \)
59 \( 1 + (-8.54 - 4.93i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (36.5 + 63.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-12.6 + 21.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 97.8iT - 5.04e3T^{2} \)
73 \( 1 + 77.0T + 5.32e3T^{2} \)
79 \( 1 + (-42.1 - 72.9i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (40.6 - 23.4i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + (32.4 + 56.1i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.66355901751606615571745302920, −11.48634382508694136627561968052, −10.56942584176074474355538697395, −9.363072675793024355035923824607, −7.932966262357156712305815644614, −7.07011977070879293773214021921, −6.14047660496434715850737220496, −4.49586828230051920637227776574, −2.66734909087611783513646715986, −0.57645239943466851861490829165, 2.73741027141420020889982905543, 4.19267157108057558566510716936, 5.59334398225314217689096986060, 6.45923638623339404036275382141, 8.475361861381515912698265787219, 9.052868796306161900615736988079, 10.32911520288144750011855801216, 11.01106948900678154606853799350, 12.33507801295175876331838410265, 13.00099466337469329658760533086

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