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Properties

Label 2-12e2-144.131-c1-0-17
Degree $2$
Conductor $144$
Sign $0.932 + 0.361i$
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  + (1.37 + 0.313i)2-s + (−0.412 − 1.68i)3-s + (1.80 + 0.864i)4-s + (0.289 − 0.0776i)5-s + (−0.0413 − 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 + (−2.23 − 0.599i)11-s + (0.710 − 3.39i)12-s + (1.60 − 0.429i)13-s + (−0.312 − 1.01i)14-s + (−0.250 − 0.455i)15-s + (2.50 + 3.11i)16-s + 6.74i·17-s + ⋯
L(s)  = 1  + (0.975 + 0.221i)2-s + (−0.238 − 0.971i)3-s + (0.901 + 0.432i)4-s + (0.129 − 0.0347i)5-s + (−0.0168 − 0.999i)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 + (−0.674 − 0.180i)11-s + (0.205 − 0.978i)12-s + (0.444 − 0.119i)13-s + (−0.0835 − 0.270i)14-s + (−0.0646 − 0.117i)15-s + (0.626 + 0.779i)16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69078 - 0.316211i\)
\(L(\frac12)\) \(\approx\) \(1.69078 - 0.316211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.313i)T \)
3 \( 1 + (0.412 + 1.68i)T \)
good5 \( 1 + (-0.289 + 0.0776i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.374 + 0.647i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.23 + 0.599i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.60 + 0.429i)T + (11.2 - 6.5i)T^{2} \)
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 \( 1 + (5.44 + 1.45i)T + (25.1 + 14.5i)T^{2} \)
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 \( 1 + (-1.89 + 7.08i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.307 - 0.531i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \)
59 \( 1 + (0.00225 + 0.00841i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.72 + 10.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.35 - 8.78i)T + (-58.0 + 33.5i)T^{2} \)
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 \( 1 + (0.353 - 1.32i)T + (-71.8 - 41.5i)T^{2} \)
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

−13.09186636103524919170845110509, −12.37532955449714182611214274293, −11.29253172027245095052613404801, −10.39374843793246288207537101183, −8.361603617326897747743715954556, −7.56343686778542948071358894149, −6.28600904827768894119665337722, −5.59391039283571654679492599052, −3.85241123412653493640607115307, −2.12535137948752662973328731355, 2.72645727321152744346519573415, 4.10159529794346048901906420524, 5.26892466771452913433525639915, 6.16675096186353543786333771887, 7.73935945803061370332765988241, 9.469173341628193747359747034544, 10.18776804418061071573925580947, 11.38803417472234654263041128177, 11.93105627305453956123702832573, 13.32231075606213348735753476343

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