Properties

Degree $2$
Conductor $144$
Sign $0.695 + 0.718i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.12 − 0.854i)2-s + (0.540 + 1.92i)4-s + (0.763 − 0.763i)5-s + 1.33·7-s + (1.03 − 2.63i)8-s + (−1.51 + 0.208i)10-s + (1.95 + 1.95i)11-s + (4.18 − 4.18i)13-s + (−1.50 − 1.14i)14-s + (−3.41 + 2.08i)16-s − 4.03i·17-s + (−4.26 − 4.26i)19-s + (1.88 + 1.05i)20-s + (−0.534 − 3.88i)22-s + 8.86i·23-s + ⋯
L(s)  = 1  + (−0.796 − 0.604i)2-s + (0.270 + 0.962i)4-s + (0.341 − 0.341i)5-s + 0.505·7-s + (0.366 − 0.930i)8-s + (−0.478 + 0.0658i)10-s + (0.590 + 0.590i)11-s + (1.16 − 1.16i)13-s + (−0.402 − 0.305i)14-s + (−0.854 + 0.520i)16-s − 0.978i·17-s + (−0.979 − 0.979i)19-s + (0.421 + 0.236i)20-s + (−0.113 − 0.827i)22-s + 1.84i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.695 + 0.718i$
Motivic weight: \(1\)
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804467 - 0.340636i\)
\(L(\frac12)\) \(\approx\) \(0.804467 - 0.340636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.12 + 0.854i)T \)
3 \( 1 \)
good5 \( 1 + (-0.763 + 0.763i)T - 5iT^{2} \)
7 \( 1 - 1.33T + 7T^{2} \)
11 \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \)
13 \( 1 + (-4.18 + 4.18i)T - 13iT^{2} \)
17 \( 1 + 4.03iT - 17T^{2} \)
19 \( 1 + (4.26 + 4.26i)T + 19iT^{2} \)
23 \( 1 - 8.86iT - 23T^{2} \)
29 \( 1 + (-1.23 - 1.23i)T + 29iT^{2} \)
31 \( 1 - 2.87iT - 31T^{2} \)
37 \( 1 + (-0.434 - 0.434i)T + 37iT^{2} \)
41 \( 1 + 7.81T + 41T^{2} \)
43 \( 1 + (5.49 - 5.49i)T - 43iT^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + (-4.06 + 4.06i)T - 53iT^{2} \)
59 \( 1 + (4.71 + 4.71i)T + 59iT^{2} \)
61 \( 1 + (-3.26 + 3.26i)T - 61iT^{2} \)
67 \( 1 + (5.44 + 5.44i)T + 67iT^{2} \)
71 \( 1 + 3.76iT - 71T^{2} \)
73 \( 1 - 10.5iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (-9.73 + 9.73i)T - 83iT^{2} \)
89 \( 1 + 1.64T + 89T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−12.94446861942873579768804166143, −11.68390651071278090029548774368, −11.00200318448600736634277141639, −9.817660570765216787105988975598, −8.951383488323182150100206686155, −7.968055480975901492152817385913, −6.74876185229146804235629505709, −5.03754162128245509075340982690, −3.36733806326437601363928003519, −1.50493045485174487212830001669, 1.80752032818315695640285654960, 4.25234419758226984170054508496, 6.11077821056900959465925483579, 6.56795340910996606679720161616, 8.313254955761947082258923069756, 8.728541671341192684329105713785, 10.21144333281829946770051724021, 10.88648653664541068857094963459, 11.97221996076481920964552251051, 13.59524550415990105324905923981

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