Properties

Degree $2$
Conductor $144$
Sign $0.445 + 0.895i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.445 + 0.895i$
Motivic weight: \(1\)
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35361 - 0.838832i\)
\(L(\frac12)\) \(\approx\) \(1.35361 - 0.838832i\)
\(L(\frac{3}{2})\) 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

−12.93284940847541405399189307670, −11.83692878391796435240340789853, −11.24916309364268471082011327030, −10.06925890591722147607370683451, −8.923885710538010418543982207923, −7.51744553197712260346154714656, −6.10645358015957432775583308745, −4.82888994961844820546655956003, −3.79017462950688121524413506771, −1.85786522019020539428296435516, 2.91351473777109356366258258133, 4.26709991156210722201910819356, 5.63090772232286782571788032644, 6.64621052386853177098683182746, 8.059316769708845201460744944031, 8.548272960770936051295246193254, 10.75248620400562165791615774241, 11.13660906649380794199119324923, 12.69550370210802336437058882068, 13.17518285154771886950654893031

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