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Properties

Label 2-12e2-36.7-c2-0-4
Degree $2$
Conductor $144$
Sign $0.999 - 0.0435i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.456 − 2.96i)3-s + (4.61 + 7.99i)5-s + (5.33 + 3.07i)7-s + (−8.58 − 2.70i)9-s + (3.70 + 2.13i)11-s + (0.869 + 1.50i)13-s + (25.8 − 10.0i)15-s + 12.3·17-s − 33.9i·19-s + (11.5 − 14.4i)21-s + (3.35 − 1.93i)23-s + (−30.1 + 52.1i)25-s + (−11.9 + 24.2i)27-s + (17.8 − 30.9i)29-s + (−38.8 + 22.4i)31-s + ⋯
L(s)  = 1  + (0.152 − 0.988i)3-s + (0.923 + 1.59i)5-s + (0.761 + 0.439i)7-s + (−0.953 − 0.300i)9-s + (0.336 + 0.194i)11-s + (0.0668 + 0.115i)13-s + (1.72 − 0.669i)15-s + 0.726·17-s − 1.78i·19-s + (0.550 − 0.685i)21-s + (0.145 − 0.0841i)23-s + (−1.20 + 2.08i)25-s + (−0.442 + 0.896i)27-s + (0.615 − 1.06i)29-s + (−1.25 + 0.723i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0435i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.0435i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.999 - 0.0435i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.79184 + 0.0390731i\)
\(L(\frac12)\) \(\approx\) \(1.79184 + 0.0390731i\)
\(L(2)\) 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 \( 1 + (-0.456 + 2.96i)T \)
good5 \( 1 + (-4.61 - 7.99i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.33 - 3.07i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.70 - 2.13i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-0.869 - 1.50i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 12.3T + 289T^{2} \)
19 \( 1 + 33.9iT - 361T^{2} \)
23 \( 1 + (-3.35 + 1.93i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-17.8 + 30.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (38.8 - 22.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 32.7T + 1.36e3T^{2} \)
41 \( 1 + (-21.8 - 37.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (33.9 + 19.5i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (39.8 + 23.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 46.3T + 2.80e3T^{2} \)
59 \( 1 + (23.2 - 13.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.4 + 40.6i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-56.9 + 32.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 96.7iT - 5.04e3T^{2} \)
73 \( 1 + 14.0T + 5.32e3T^{2} \)
79 \( 1 + (34.3 + 19.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (81.7 + 47.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 81.8T + 7.92e3T^{2} \)
97 \( 1 + (7.99 - 13.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.05728364700432303811339813711, −11.69661153873662080887914995706, −11.06735515584772010936085052300, −9.819432988368016098369892472073, −8.606334780195391236739101174357, −7.25023643093122474692391412584, −6.59152853389184431359088246717, −5.41525695758708801637139498472, −3.01659970007905456229927701724, −1.91727755357697426061247161673, 1.50359759019870865259166482133, 3.86363145228645716175168258560, 5.05041716409600772187036514376, 5.77936079374371851036674663043, 7.991015094035829036982351465190, 8.805093853584310565568551065001, 9.726025420118463350131954785228, 10.57322134948996654327211890404, 11.89523012606896837106929524185, 12.86497078817372100109372862321

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