Properties

Degree $2$
Conductor $144$
Sign $-0.825 + 0.564i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.263 − 1.38i)2-s + (−1.86 − 0.731i)4-s + (−2.63 − 2.63i)5-s − 0.207·7-s + (−1.50 + 2.39i)8-s + (−4.35 + 2.96i)10-s + (3.66 − 3.66i)11-s + (0.255 + 0.255i)13-s + (−0.0545 + 0.287i)14-s + (2.93 + 2.72i)16-s + 0.654i·17-s + (4.46 − 4.46i)19-s + (2.97 + 6.82i)20-s + (−4.13 − 6.06i)22-s + 3.48i·23-s + ⋯
L(s)  = 1  + (0.186 − 0.982i)2-s + (−0.930 − 0.365i)4-s + (−1.17 − 1.17i)5-s − 0.0783·7-s + (−0.532 + 0.846i)8-s + (−1.37 + 0.937i)10-s + (1.10 − 1.10i)11-s + (0.0708 + 0.0708i)13-s + (−0.0145 + 0.0769i)14-s + (0.732 + 0.680i)16-s + 0.158i·17-s + (1.02 − 1.02i)19-s + (0.665 + 1.52i)20-s + (−0.880 − 1.29i)22-s + 0.727i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 + 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.825 + 0.564i$
Motivic weight: \(1\)
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.825 + 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262723 - 0.850442i\)
\(L(\frac12)\) \(\approx\) \(0.262723 - 0.850442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.263 + 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (2.63 + 2.63i)T + 5iT^{2} \)
7 \( 1 + 0.207T + 7T^{2} \)
11 \( 1 + (-3.66 + 3.66i)T - 11iT^{2} \)
13 \( 1 + (-0.255 - 0.255i)T + 13iT^{2} \)
17 \( 1 - 0.654iT - 17T^{2} \)
19 \( 1 + (-4.46 + 4.46i)T - 19iT^{2} \)
23 \( 1 - 3.48iT - 23T^{2} \)
29 \( 1 + (4.33 - 4.33i)T - 29iT^{2} \)
31 \( 1 + 6.16iT - 31T^{2} \)
37 \( 1 + (-4.39 + 4.39i)T - 37iT^{2} \)
41 \( 1 + 0.0684T + 41T^{2} \)
43 \( 1 + (-5.65 - 5.65i)T + 43iT^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 + (1.51 + 1.51i)T + 53iT^{2} \)
59 \( 1 + (2.53 - 2.53i)T - 59iT^{2} \)
61 \( 1 + (5.46 + 5.46i)T + 61iT^{2} \)
67 \( 1 + (4.77 - 4.77i)T - 67iT^{2} \)
71 \( 1 - 5.94iT - 71T^{2} \)
73 \( 1 - 6.93iT - 73T^{2} \)
79 \( 1 - 4.72iT - 79T^{2} \)
83 \( 1 + (4.32 + 4.32i)T + 83iT^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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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

−12.56979593147511468781615052168, −11.50725610609870841347331562205, −11.25208306528826584636285412213, −9.374436339431258801537605937290, −8.872509543714219310333028264859, −7.67922303281608873696673701054, −5.69638496590781200212928219067, −4.40401190421874285081992318574, −3.44140390952219715933913615276, −0.947942240431247221091139007976, 3.41999606282690611058903004178, 4.43790516852416569830323569940, 6.20736471539957289080660434228, 7.18600428155352425538231939579, 7.81994444576428475033281022055, 9.231984730420499862872006278112, 10.36453706529881412399772645123, 11.77268540269220008393867429733, 12.38661639841289694926603361956, 13.93711257158613464037167889787

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