Properties

Label 2-12e2-144.59-c1-0-18
Degree $2$
Conductor $144$
Sign $0.109 + 0.993i$
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  + (0.417 − 1.35i)2-s + (1.68 − 0.412i)3-s + (−1.65 − 1.12i)4-s + (0.0776 + 0.289i)5-s + (0.145 − 2.44i)6-s + (−0.374 − 0.647i)7-s + (−2.21 + 1.75i)8-s + (2.65 − 1.38i)9-s + (0.424 + 0.0162i)10-s + (−0.599 + 2.23i)11-s + (−3.24 − 1.21i)12-s + (−0.429 − 1.60i)13-s + (−1.03 + 0.234i)14-s + (0.250 + 0.455i)15-s + (1.44 + 3.72i)16-s + 6.74i·17-s + ⋯
L(s)  = 1  + (0.295 − 0.955i)2-s + (0.971 − 0.238i)3-s + (−0.825 − 0.564i)4-s + (0.0347 + 0.129i)5-s + (0.0595 − 0.998i)6-s + (−0.141 − 0.244i)7-s + (−0.783 + 0.621i)8-s + (0.886 − 0.462i)9-s + (0.134 + 0.00513i)10-s + (−0.180 + 0.674i)11-s + (−0.936 − 0.351i)12-s + (−0.119 − 0.444i)13-s + (−0.275 + 0.0626i)14-s + (0.0646 + 0.117i)15-s + (0.362 + 0.932i)16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.109 + 0.993i$
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: \((2,\ 144,\ (\ :1/2),\ 0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12208 - 1.00524i\)
\(L(\frac12)\) \(\approx\) \(1.12208 - 1.00524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.417 + 1.35i)T \)
3 \( 1 + (-1.68 + 0.412i)T \)
good5 \( 1 + (-0.0776 - 0.289i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.374 + 0.647i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.599 - 2.23i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.429 + 1.60i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.74iT - 17T^{2} \)
19 \( 1 + (-0.621 - 0.621i)T + 19iT^{2} \)
23 \( 1 + (6.06 + 3.49i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.45 - 5.44i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.13 + 1.81i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.74 - 6.74i)T + 37iT^{2} \)
41 \( 1 + (-1.39 + 2.41i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.08 + 1.89i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.307 + 0.531i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.68 - 2.68i)T - 53iT^{2} \)
59 \( 1 + (0.00841 - 0.00225i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.1 + 2.72i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (8.78 - 2.35i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + 8.17iT - 73T^{2} \)
79 \( 1 + (-7.67 + 4.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.32 + 0.353i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 + (4.62 + 8.00i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.79027104894566039108578455774, −12.22660685343324179071156829499, −10.58429748076798345583955724938, −10.03402414828693165573202313947, −8.821574165922900545693839916062, −7.84486209372750291637146369528, −6.28064090533603253596256201413, −4.50644387270538857009212554464, −3.33406575042206971061104331918, −1.89396166992104960453161722221, 2.96061806072062111371755827956, 4.33935653786400642823400335734, 5.62325464928213359991202752736, 7.10434353600315564927218685836, 8.000659614302463695553627550152, 9.106501236124203490239883803660, 9.702615428104791910132034478770, 11.46879230932796747930294331046, 12.75705222400579554255776139920, 13.69708766966016227799513850576

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