Properties

Label 2-12e2-144.131-c1-0-15
Degree $2$
Conductor $144$
Sign $-0.567 + 0.823i$
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  + (−0.990 − 1.00i)2-s + (−0.800 − 1.53i)3-s + (−0.0363 + 1.99i)4-s + (3.73 − 1.00i)5-s + (−0.757 + 2.32i)6-s + (−1.68 − 2.91i)7-s + (2.05 − 1.94i)8-s + (−1.71 + 2.45i)9-s + (−4.70 − 2.77i)10-s + (0.211 + 0.0566i)11-s + (3.10 − 1.54i)12-s + (−2.71 + 0.727i)13-s + (−1.27 + 4.58i)14-s + (−4.52 − 4.93i)15-s + (−3.99 − 0.145i)16-s − 4.23i·17-s + ⋯
L(s)  = 1  + (−0.700 − 0.713i)2-s + (−0.461 − 0.886i)3-s + (−0.0181 + 0.999i)4-s + (1.67 − 0.447i)5-s + (−0.309 + 0.951i)6-s + (−0.635 − 1.10i)7-s + (0.726 − 0.687i)8-s + (−0.573 + 0.819i)9-s + (−1.48 − 0.878i)10-s + (0.0637 + 0.0170i)11-s + (0.895 − 0.445i)12-s + (−0.753 + 0.201i)13-s + (−0.340 + 1.22i)14-s + (−1.16 − 1.27i)15-s + (−0.999 − 0.0363i)16-s − 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.567 + 0.823i$
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: \((2,\ 144,\ (\ :1/2),\ -0.567 + 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.367156 - 0.698721i\)
\(L(\frac12)\) \(\approx\) \(0.367156 - 0.698721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.990 + 1.00i)T \)
3 \( 1 + (0.800 + 1.53i)T \)
good5 \( 1 + (-3.73 + 1.00i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.68 + 2.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.211 - 0.0566i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.71 - 0.727i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 4.23iT - 17T^{2} \)
19 \( 1 + (1.12 - 1.12i)T - 19iT^{2} \)
23 \( 1 + (-3.33 - 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.03 - 0.545i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-7.21 - 4.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.66 - 2.66i)T - 37iT^{2} \)
41 \( 1 + (-1.70 + 2.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.25 - 4.68i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (2.34 + 4.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.58 - 7.58i)T + 53iT^{2} \)
59 \( 1 + (1.43 + 5.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.33 - 8.69i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.38 - 5.17i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.53iT - 71T^{2} \)
73 \( 1 + 3.22iT - 73T^{2} \)
79 \( 1 + (-4.98 + 2.87i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.20 + 4.50i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 2.96T + 89T^{2} \)
97 \( 1 + (7.63 + 13.2i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.76639007245847927115773692263, −11.81352122578504721520084444570, −10.46606266457231588115009601307, −9.883978839606915034771189367312, −8.824479635884355203399544082721, −7.31664731849060580306718477400, −6.51543572122966010298790242501, −4.93776024989797545295092453686, −2.64577141399610779011878069013, −1.14321988557430552495786442776, 2.50278069165753680827078407150, 5.06579228382206163992886604698, 5.99046787237029157077738666803, 6.56767973727411199032656195356, 8.620814335613272308249785934638, 9.506932351722035447443499216086, 10.04282481187348562580982141647, 10.91847226723341408283714930385, 12.44191311139510994085624680684, 13.71721557174587199436119827128

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