Properties

Label 2-12e2-144.85-c1-0-1
Degree $2$
Conductor $144$
Sign $0.0807 - 0.996i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 − 0.161i)2-s + (−1.68 − 0.388i)3-s + (1.94 + 0.452i)4-s + (−0.846 + 0.226i)5-s + (2.30 + 0.818i)6-s + (−0.567 + 0.327i)7-s + (−2.66 − 0.950i)8-s + (2.69 + 1.31i)9-s + (1.22 − 0.182i)10-s + (−1.54 + 5.75i)11-s + (−3.11 − 1.52i)12-s + (1.19 + 4.44i)13-s + (0.849 − 0.368i)14-s + (1.51 − 0.0535i)15-s + (3.58 + 1.76i)16-s + 2.75·17-s + ⋯
L(s)  = 1  + (−0.993 − 0.113i)2-s + (−0.974 − 0.224i)3-s + (0.974 + 0.226i)4-s + (−0.378 + 0.101i)5-s + (0.942 + 0.334i)6-s + (−0.214 + 0.123i)7-s + (−0.941 − 0.335i)8-s + (0.899 + 0.437i)9-s + (0.387 − 0.0576i)10-s + (−0.464 + 1.73i)11-s + (−0.898 − 0.439i)12-s + (0.330 + 1.23i)13-s + (0.227 − 0.0985i)14-s + (0.391 − 0.0138i)15-s + (0.897 + 0.441i)16-s + 0.668·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0807 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0807 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.0807 - 0.996i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.0807 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282398 + 0.260454i\)
\(L(\frac12)\) \(\approx\) \(0.282398 + 0.260454i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.161i)T \)
3 \( 1 + (1.68 + 0.388i)T \)
good5 \( 1 + (0.846 - 0.226i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.567 - 0.327i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.54 - 5.75i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.19 - 4.44i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 + (1.73 - 1.73i)T - 19iT^{2} \)
23 \( 1 + (3.50 + 2.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.47 - 0.662i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-2.08 + 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 + 4.30i)T + 37iT^{2} \)
41 \( 1 + (-6.15 - 3.55i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.225 + 0.841i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.65 - 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.64 + 7.64i)T + 53iT^{2} \)
59 \( 1 + (-6.83 + 1.83i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.77 + 1.01i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.11 - 11.6i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 + 0.656iT - 73T^{2} \)
79 \( 1 + (8.16 + 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.36 - 1.43i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 5.11iT - 89T^{2} \)
97 \( 1 + (-3.05 - 5.29i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.77030770996421948149500314645, −12.13371498521835251183896442565, −11.30836686722004260026193093146, −10.21874984186800219712302143051, −9.503738463473140825308672638458, −7.898598655296024886586056844056, −7.07971705868213156127691141306, −6.04086210856138366358574645933, −4.30189706250965551485811576163, −1.93275231515143715651827467664, 0.58700115170336466262120710353, 3.37325627930849963112822428558, 5.49059648948558774356182173374, 6.28057595039859116488532054690, 7.72385189194862174240737633197, 8.591817712453644917216950901104, 10.05976077005727402706131988829, 10.69724287027280646997570516564, 11.56200380296616518586902439079, 12.49560389689542898415303270535

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