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Properties

Label 2-12e2-12.11-c1-0-1
Degree $2$
Conductor $144$
Sign $0.577 + 0.816i$
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  − 4.24i·5-s + 4·13-s + 4.24i·17-s − 12.9·25-s + 4.24i·29-s + 2·37-s + 12.7i·41-s + 7·49-s − 12.7i·53-s − 10·61-s − 16.9i·65-s + 16·73-s + 17.9·85-s − 4.24i·89-s − 8·97-s + ⋯
L(s)  = 1  − 1.89i·5-s + 1.10·13-s + 1.02i·17-s − 2.59·25-s + 0.787i·29-s + 0.328·37-s + 1.98i·41-s + 49-s − 1.74i·53-s − 1.28·61-s − 2.10i·65-s + 1.87·73-s + 1.95·85-s − 0.449i·89-s − 0.812·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.985446 - 0.510104i\)
\(L(\frac12)\) \(\approx\) \(0.985446 - 0.510104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 4.24iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 4.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.24iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 16T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−12.94860897294875244855465314339, −12.16288692014256542391037416544, −11.01743364068965368102355449919, −9.656087274818344088634370141675, −8.682770101656673594743778626989, −8.047594690926185793623914759732, −6.20960679757941866437092235792, −5.08339899657969180788829960218, −3.91618276927594421014409614953, −1.37578877750920465414474926871, 2.59850145117782529263796106565, 3.82882281227794167043136970311, 5.83724029182699751068136356638, 6.82380403811947813379485228639, 7.74240346995427529361484942778, 9.256047475851871712058476339581, 10.45561598678540356209631569305, 11.06515689393387037656719208575, 12.01926027725275422018720232818, 13.66087435228723288266073593815

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