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Properties

Label 2-12e2-144.133-c1-0-9
Degree $2$
Conductor $144$
Sign $0.999 + 0.00740i$
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  + (1.30 − 0.550i)2-s + (−1.21 + 1.22i)3-s + (1.39 − 1.43i)4-s + (−0.0468 + 0.174i)5-s + (−0.912 + 2.27i)6-s + (4.04 + 2.33i)7-s + (1.02 − 2.63i)8-s + (−0.0241 − 2.99i)9-s + (0.0351 + 0.253i)10-s + (−0.598 + 0.160i)11-s + (0.0627 + 3.46i)12-s + (−4.41 − 1.18i)13-s + (6.55 + 0.816i)14-s + (−0.157 − 0.270i)15-s + (−0.112 − 3.99i)16-s − 4.34·17-s + ⋯
L(s)  = 1  + (0.921 − 0.389i)2-s + (−0.704 + 0.709i)3-s + (0.697 − 0.716i)4-s + (−0.0209 + 0.0781i)5-s + (−0.372 + 0.928i)6-s + (1.52 + 0.883i)7-s + (0.363 − 0.931i)8-s + (−0.00806 − 0.999i)9-s + (0.0111 + 0.0801i)10-s + (−0.180 + 0.0483i)11-s + (0.0181 + 0.999i)12-s + (−1.22 − 0.327i)13-s + (1.75 + 0.218i)14-s + (−0.0407 − 0.0698i)15-s + (−0.0281 − 0.999i)16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 + 0.00740i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.999 + 0.00740i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57736 - 0.00584189i\)
\(L(\frac12)\) \(\approx\) \(1.57736 - 0.00584189i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.550i)T \)
3 \( 1 + (1.21 - 1.22i)T \)
good5 \( 1 + (0.0468 - 0.174i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-4.04 - 2.33i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.598 - 0.160i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.41 + 1.18i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 + (3.86 - 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.31 + 8.64i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (2.25 + 3.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.79 - 2.79i)T + 37iT^{2} \)
41 \( 1 + (-3.67 + 2.12i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.0131 - 0.00351i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (1.17 - 2.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.519 + 0.519i)T + 53iT^{2} \)
59 \( 1 + (2.95 - 11.0i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.588 - 2.19i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-7.04 - 1.88i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.55iT - 71T^{2} \)
73 \( 1 + 2.92iT - 73T^{2} \)
79 \( 1 + (-1.45 + 2.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.99 - 7.44i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.18iT - 89T^{2} \)
97 \( 1 + (-8.03 + 13.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.87954378993050905782257114099, −11.84366945716299976523407466131, −11.37663297117618852698862547668, −10.40890897228331980456503602961, −9.267162933548278284646144893396, −7.66010837273689396981938343316, −6.04309828302893764346248748506, −5.11484786636068878271642663689, −4.29541722014113673135693635292, −2.30662752398928025661270380539, 2.06713649045134293165188693029, 4.50128645339276521769818344427, 5.11906212096217091705261345568, 6.70533612440639539807757082415, 7.43618634346110570970907194521, 8.416210636972137644528021066069, 10.68971146899647538609201255257, 11.22386597053567400382754203918, 12.27129845839619483392344482032, 13.04642067087922961059518050148

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