# Properties

 Degree $2$ Conductor $144$ Sign $0.745 - 0.666i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (−0.957 + 1.04i)2-s + (−0.167 − 1.99i)4-s + (0.236 − 0.236i)5-s + 3.27·7-s + (2.23 + 1.73i)8-s + (0.0197 + 0.472i)10-s + (2.58 + 2.58i)11-s + (−1.70 + 1.70i)13-s + (−3.13 + 3.41i)14-s + (−3.94 + 0.665i)16-s − 7.05i·17-s + (3.04 + 3.04i)19-s + (−0.510 − 0.431i)20-s + (−5.16 + 0.215i)22-s + 1.47i·23-s + ⋯
 L(s)  = 1 + (−0.676 + 0.736i)2-s + (−0.0835 − 0.996i)4-s + (0.105 − 0.105i)5-s + 1.23·7-s + (0.790 + 0.613i)8-s + (0.00624 + 0.149i)10-s + (0.778 + 0.778i)11-s + (−0.473 + 0.473i)13-s + (−0.838 + 0.912i)14-s + (−0.986 + 0.166i)16-s − 1.71i·17-s + (0.697 + 0.697i)19-s + (−0.114 − 0.0964i)20-s + (−1.10 + 0.0460i)22-s + 0.307i·23-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$144$$    =    $$2^{4} \cdot 3^{2}$$ Sign: $0.745 - 0.666i$ Motivic weight: $$1$$ Character: $\chi_{144} (107, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 144,\ (\ :1/2),\ 0.745 - 0.666i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.848376 + 0.324119i$$ $$L(\frac12)$$ $$\approx$$ $$0.848376 + 0.324119i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.957 - 1.04i)T$$
3 $$1$$
good5 $$1 + (-0.236 + 0.236i)T - 5iT^{2}$$
7 $$1 - 3.27T + 7T^{2}$$
11 $$1 + (-2.58 - 2.58i)T + 11iT^{2}$$
13 $$1 + (1.70 - 1.70i)T - 13iT^{2}$$
17 $$1 + 7.05iT - 17T^{2}$$
19 $$1 + (-3.04 - 3.04i)T + 19iT^{2}$$
23 $$1 - 1.47iT - 23T^{2}$$
29 $$1 + (2.98 + 2.98i)T + 29iT^{2}$$
31 $$1 + 8.02iT - 31T^{2}$$
37 $$1 + (7.93 + 7.93i)T + 37iT^{2}$$
41 $$1 - 2.22T + 41T^{2}$$
43 $$1 + (4.61 - 4.61i)T - 43iT^{2}$$
47 $$1 + 7.13T + 47T^{2}$$
53 $$1 + (5.81 - 5.81i)T - 53iT^{2}$$
59 $$1 + (-7.46 - 7.46i)T + 59iT^{2}$$
61 $$1 + (4.04 - 4.04i)T - 61iT^{2}$$
67 $$1 + (2.90 + 2.90i)T + 67iT^{2}$$
71 $$1 - 1.02iT - 71T^{2}$$
73 $$1 + 4.08iT - 73T^{2}$$
79 $$1 + 5.36iT - 79T^{2}$$
83 $$1 + (3.93 - 3.93i)T - 83iT^{2}$$
89 $$1 + 2.35T + 89T^{2}$$
97 $$1 - 9.97T + 97T^{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

−13.60991336916697375824256459810, −11.87542305701360114167001028699, −11.23819679833684546955256862785, −9.734564246892325468804344070013, −9.177774931834446767647892342932, −7.75192844161350779332853086773, −7.13986512200461492705201634079, −5.54624297086259203620664138904, −4.54150835306952303932876412721, −1.73326008034501168371744012335, 1.60107677076249520353303756797, 3.42577724117731872585422721826, 4.94052685595253815119049126499, 6.73181547449944922272193093975, 8.158000561862939549604258621448, 8.681425322338248942052411601031, 10.10698114669409711629479069664, 10.93911078291367855116840750000, 11.76225391784834443935327308666, 12.70578779572280557897486233550