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: $\chi_{144} (31, \cdot )$
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

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

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