Properties

Degree $2$
Conductor $144$
Sign $-0.537 - 0.843i$
Motivic weight $2$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.537 - 0.843i$
Motivic weight: \(2\)
Character: $\chi_{144} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.537 - 0.843i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.628693 + 1.14591i\)
\(L(\frac12)\) \(\approx\) \(0.628693 + 1.14591i\)
\(L(2)\) 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

−13.56839672280234191243500143409, −11.98841326221543560365604245677, −10.79583110136142655983404326054, −10.08740699254254703480522363217, −9.686404584841958779243584018201, −7.893363551150228800884641510526, −6.41081856232672334286942740709, −5.77055316125661783569707657419, −3.82408886824285930302862034991, −2.76542418881860674210321878122, 0.898585450528155957819710722484, 2.54212287683794237982540762268, 4.99350331169179216802146706196, 5.82530849102904714351119061198, 7.03357197289673777925829692219, 8.552677892063699416170965133692, 9.114016008399341321143172426941, 10.37676218749022329835850761122, 12.05232878115219662778458841360, 12.55547153832009357504493785514

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