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Properties

Label 2-12e2-144.11-c1-0-10
Degree $2$
Conductor $144$
Sign $0.999 - 0.00151i$
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  + (−0.354 + 1.36i)2-s + (−0.874 − 1.49i)3-s + (−1.74 − 0.970i)4-s + (1.76 + 0.473i)5-s + (2.35 − 0.667i)6-s + (1.40 − 2.43i)7-s + (1.94 − 2.04i)8-s + (−1.46 + 2.61i)9-s + (−1.27 + 2.25i)10-s + (5.79 − 1.55i)11-s + (0.0781 + 3.46i)12-s + (−1.10 − 0.296i)13-s + (2.83 + 2.78i)14-s + (−0.837 − 3.05i)15-s + (2.11 + 3.39i)16-s + 0.699i·17-s + ⋯
L(s)  = 1  + (−0.250 + 0.968i)2-s + (−0.505 − 0.863i)3-s + (−0.874 − 0.485i)4-s + (0.789 + 0.211i)5-s + (0.962 − 0.272i)6-s + (0.531 − 0.920i)7-s + (0.689 − 0.724i)8-s + (−0.489 + 0.871i)9-s + (−0.403 + 0.711i)10-s + (1.74 − 0.468i)11-s + (0.0225 + 0.999i)12-s + (−0.307 − 0.0823i)13-s + (0.757 + 0.745i)14-s + (−0.216 − 0.788i)15-s + (0.528 + 0.848i)16-s + 0.169i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.00151i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.999 - 0.00151i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933845 + 0.000708658i\)
\(L(\frac12)\) \(\approx\) \(0.933845 + 0.000708658i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.354 - 1.36i)T \)
3 \( 1 + (0.874 + 1.49i)T \)
good5 \( 1 + (-1.76 - 0.473i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.40 + 2.43i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.79 + 1.55i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.10 + 0.296i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 0.699iT - 17T^{2} \)
19 \( 1 + (2.01 + 2.01i)T + 19iT^{2} \)
23 \( 1 + (4.91 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.41 + 1.45i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.15 - 2.97i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 + (-1.55 - 2.70i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.90 + 7.09i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.83 - 4.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.45 - 4.45i)T - 53iT^{2} \)
59 \( 1 + (3.65 - 13.6i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.54 - 13.2i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.45 + 12.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.21iT - 71T^{2} \)
73 \( 1 + 3.75iT - 73T^{2} \)
79 \( 1 + (2.96 + 1.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.308 - 1.15i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 0.391T + 89T^{2} \)
97 \( 1 + (0.875 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.61586938248255682417643574328, −12.19329807703532933215792195046, −11.00527476962665027925772731401, −9.939820249412666685414288929834, −8.707594156575562214700004672664, −7.55206891942666942287505102884, −6.59576415486298559481273159561, −5.86476758351917166446329461827, −4.34056021144612118595793980574, −1.37347175911350770897544187414, 1.96082088071371502367803087194, 3.91822360936298897742637171809, 5.04782825041708865365939733359, 6.26884091332593156658231852940, 8.429195004414463941950543648031, 9.396044605043260166113713055456, 9.859740269984044781564237457439, 11.17843494610297855831153805666, 11.92212004884593709493409683002, 12.63782224988133879006028844464

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