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Properties

Label 2-12e2-48.5-c2-0-7
Degree $2$
Conductor $144$
Sign $0.820 + 0.571i$
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  + (−1.95 − 0.412i)2-s + (3.66 + 1.61i)4-s + (5.52 − 5.52i)5-s + 7.79i·7-s + (−6.49 − 4.66i)8-s + (−13.0 + 8.53i)10-s + (−2.48 + 2.48i)11-s + (13.3 − 13.3i)13-s + (3.21 − 15.2i)14-s + (10.7 + 11.8i)16-s + 5.86i·17-s + (18.5 − 18.5i)19-s + (29.1 − 11.3i)20-s + (5.88 − 3.83i)22-s + 34.3·23-s + ⋯
L(s)  = 1  + (−0.978 − 0.206i)2-s + (0.915 + 0.403i)4-s + (1.10 − 1.10i)5-s + 1.11i·7-s + (−0.812 − 0.583i)8-s + (−1.30 + 0.853i)10-s + (−0.225 + 0.225i)11-s + (1.02 − 1.02i)13-s + (0.229 − 1.08i)14-s + (0.674 + 0.738i)16-s + 0.344i·17-s + (0.977 − 0.977i)19-s + (1.45 − 0.565i)20-s + (0.267 − 0.174i)22-s + 1.49·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.820 + 0.571i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.820 + 0.571i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10223 - 0.345661i\)
\(L(\frac12)\) \(\approx\) \(1.10223 - 0.345661i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.95 + 0.412i)T \)
3 \( 1 \)
good5 \( 1 + (-5.52 + 5.52i)T - 25iT^{2} \)
7 \( 1 - 7.79iT - 49T^{2} \)
11 \( 1 + (2.48 - 2.48i)T - 121iT^{2} \)
13 \( 1 + (-13.3 + 13.3i)T - 169iT^{2} \)
17 \( 1 - 5.86iT - 289T^{2} \)
19 \( 1 + (-18.5 + 18.5i)T - 361iT^{2} \)
23 \( 1 - 34.3T + 529T^{2} \)
29 \( 1 + (21.4 + 21.4i)T + 841iT^{2} \)
31 \( 1 + 30.6T + 961T^{2} \)
37 \( 1 + (-30.3 - 30.3i)T + 1.36e3iT^{2} \)
41 \( 1 + 3.12T + 1.68e3T^{2} \)
43 \( 1 + (9.94 + 9.94i)T + 1.84e3iT^{2} \)
47 \( 1 - 38.4iT - 2.20e3T^{2} \)
53 \( 1 + (61.1 - 61.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (2.98 - 2.98i)T - 3.48e3iT^{2} \)
61 \( 1 + (-3.88 + 3.88i)T - 3.72e3iT^{2} \)
67 \( 1 + (47.0 - 47.0i)T - 4.48e3iT^{2} \)
71 \( 1 + 97.5T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 96.8T + 6.24e3T^{2} \)
83 \( 1 + (-88.8 - 88.8i)T + 6.88e3iT^{2} \)
89 \( 1 - 54.8T + 7.92e3T^{2} \)
97 \( 1 + 5.00T + 9.40e3T^{2} \)
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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

−12.84625857248501113435153627901, −11.65329429376346436404751292526, −10.56839491424148304420564538643, −9.263760899962126763589986614447, −9.020097087572212296479702189333, −7.81029160617129309101983880915, −6.12672886274042993037113354605, −5.26944571550789621996315642046, −2.80727906458156412662102955476, −1.27329954538788297515881374929, 1.56452464155128907948263613345, 3.30690379252216239232968008946, 5.67170098276846277264480061450, 6.76189580239368697578115013166, 7.44241862268611732699599910983, 9.020494976504872746033218107708, 9.894871447696418893492212543448, 10.79856144611116127144373774894, 11.33035028022214516278994470287, 13.21309051085844221933837287234

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