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: $\chi_{144} (35, \cdot )$
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

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

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