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Properties

Label 2-12e2-144.85-c1-0-10
Degree $2$
Conductor $144$
Sign $0.852 - 0.523i$
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  + (1.30 + 0.550i)2-s + (−1.71 − 0.215i)3-s + (1.39 + 1.43i)4-s + (2.73 − 0.733i)5-s + (−2.12 − 1.22i)6-s + (−1.14 + 0.660i)7-s + (1.02 + 2.63i)8-s + (2.90 + 0.740i)9-s + (3.96 + 0.551i)10-s + (−0.343 + 1.28i)11-s + (−2.08 − 2.76i)12-s + (−0.902 − 3.36i)13-s + (−1.85 + 0.230i)14-s + (−4.86 + 0.670i)15-s + (−0.114 + 3.99i)16-s − 7.60·17-s + ⋯
L(s)  = 1  + (0.921 + 0.389i)2-s + (−0.992 − 0.124i)3-s + (0.696 + 0.717i)4-s + (1.22 − 0.328i)5-s + (−0.865 − 0.500i)6-s + (−0.432 + 0.249i)7-s + (0.362 + 0.931i)8-s + (0.969 + 0.246i)9-s + (1.25 + 0.174i)10-s + (−0.103 + 0.387i)11-s + (−0.602 − 0.798i)12-s + (−0.250 − 0.933i)13-s + (−0.495 + 0.0616i)14-s + (−1.25 + 0.173i)15-s + (−0.0285 + 0.999i)16-s − 1.84·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48871 + 0.420788i\)
\(L(\frac12)\) \(\approx\) \(1.48871 + 0.420788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 - 0.550i)T \)
3 \( 1 + (1.71 + 0.215i)T \)
good5 \( 1 + (-2.73 + 0.733i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.14 - 0.660i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.343 - 1.28i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.902 + 3.36i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 7.60T + 17T^{2} \)
19 \( 1 + (-4.32 + 4.32i)T - 19iT^{2} \)
23 \( 1 + (3.46 + 1.99i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.54 + 0.950i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.569 + 0.985i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.26 - 2.26i)T + 37iT^{2} \)
41 \( 1 + (1.42 + 0.821i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.65 + 6.17i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.58 - 7.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.72 - 7.72i)T + 53iT^{2} \)
59 \( 1 + (4.80 - 1.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.92 - 2.66i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.73 + 13.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.87iT - 71T^{2} \)
73 \( 1 - 0.577iT - 73T^{2} \)
79 \( 1 + (-0.716 - 1.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.30 - 0.885i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 16.2iT - 89T^{2} \)
97 \( 1 + (0.648 + 1.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.15950483563675894483232471360, −12.50646227122345720579407998470, −11.38343867340186419922368696501, −10.33377377008540162757438247970, −9.151835774892466275849433067128, −7.40447510471955977626432579760, −6.32654093031303421178444078711, −5.55879071840845324281805924698, −4.56034203700441833007330183689, −2.36356129126355528343987099324, 1.99136859263835728203201551101, 3.95049485879235934029510843566, 5.33667795188727682915663124465, 6.23300866151011267101689896461, 6.98860649977823867252596507287, 9.478408379926335612093683988475, 10.15859679896466562374856108679, 11.14103301658640260913526297787, 11.94801432585050195027895001663, 13.19250192390503179047637211074

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