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Properties

Label 2-12e2-144.13-c1-0-11
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: $\chi_{144} (13, \cdot )$
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

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

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