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Properties

Label 2-12e2-144.59-c1-0-10
Degree $2$
Conductor $144$
Sign $-0.333 + 0.942i$
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  + (−1.09 − 0.899i)2-s + (−1.21 + 1.23i)3-s + (0.383 + 1.96i)4-s + (−0.726 − 2.70i)5-s + (2.43 − 0.254i)6-s + (−0.00424 − 0.00735i)7-s + (1.34 − 2.48i)8-s + (−0.0469 − 2.99i)9-s + (−1.64 + 3.61i)10-s + (0.804 − 3.00i)11-s + (−2.88 − 1.91i)12-s + (−1.72 − 6.42i)13-s + (−0.00197 + 0.0118i)14-s + (4.22 + 2.39i)15-s + (−3.70 + 1.50i)16-s + 3.37i·17-s + ⋯
L(s)  = 1  + (−0.771 − 0.635i)2-s + (−0.701 + 0.712i)3-s + (0.191 + 0.981i)4-s + (−0.324 − 1.21i)5-s + (0.994 − 0.104i)6-s + (−0.00160 − 0.00277i)7-s + (0.476 − 0.879i)8-s + (−0.0156 − 0.999i)9-s + (−0.519 + 1.14i)10-s + (0.242 − 0.904i)11-s + (−0.833 − 0.552i)12-s + (−0.477 − 1.78i)13-s + (−0.000528 + 0.00316i)14-s + (1.09 + 0.618i)15-s + (−0.926 + 0.376i)16-s + 0.819i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281340 - 0.398152i\)
\(L(\frac12)\) \(\approx\) \(0.281340 - 0.398152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.09 + 0.899i)T \)
3 \( 1 + (1.21 - 1.23i)T \)
good5 \( 1 + (0.726 + 2.70i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.00424 + 0.00735i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.804 + 3.00i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.72 + 6.42i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.37iT - 17T^{2} \)
19 \( 1 + (-1.35 - 1.35i)T + 19iT^{2} \)
23 \( 1 + (5.38 + 3.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.878 + 3.27i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-6.70 - 3.86i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.769 - 0.769i)T + 37iT^{2} \)
41 \( 1 + (2.64 - 4.58i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.99 + 1.60i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0955 + 0.165i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.750 - 0.750i)T - 53iT^{2} \)
59 \( 1 + (9.08 - 2.43i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-9.85 - 2.64i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (7.77 - 2.08i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.02iT - 71T^{2} \)
73 \( 1 + 7.30iT - 73T^{2} \)
79 \( 1 + (-4.36 + 2.51i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.5 - 3.35i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + (-1.64 - 2.84i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.32660257376451632401086225431, −11.85577519653603966452015474998, −10.55078285053607828484268625246, −9.965323418192064658165584675374, −8.642039325865226009313184923904, −8.050301272052564070995174462120, −6.09013506437391052235670793866, −4.74302912485109059767867875025, −3.43235699726277995030531137475, −0.68907210192245372336824273432, 2.10137390828319798802481199637, 4.75085134633994741833770841456, 6.36537046092574656714349435042, 7.01019744920844915772258380391, 7.69005001003387697919900964240, 9.346716509541023265392256956439, 10.30851678117903258073895040638, 11.47440927005813299019027794251, 11.91373187400257293486436825010, 13.77613751805715985785699160589

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