Properties

Label 2-12e2-144.59-c1-0-6
Degree $2$
Conductor $144$
Sign $0.999 - 0.0303i$
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  + (−0.725 − 1.21i)2-s + (1.33 + 1.10i)3-s + (−0.946 + 1.76i)4-s + (0.178 + 0.664i)5-s + (0.372 − 2.42i)6-s + (0.645 + 1.11i)7-s + (2.82 − 0.129i)8-s + (0.559 + 2.94i)9-s + (0.677 − 0.698i)10-s + (0.860 − 3.21i)11-s + (−3.20 + 1.30i)12-s + (1.27 + 4.74i)13-s + (0.888 − 1.59i)14-s + (−0.496 + 1.08i)15-s + (−2.20 − 3.33i)16-s − 5.58i·17-s + ⋯
L(s)  = 1  + (−0.513 − 0.858i)2-s + (0.770 + 0.637i)3-s + (−0.473 + 0.880i)4-s + (0.0796 + 0.297i)5-s + (0.152 − 0.988i)6-s + (0.244 + 0.422i)7-s + (0.998 − 0.0456i)8-s + (0.186 + 0.982i)9-s + (0.214 − 0.220i)10-s + (0.259 − 0.968i)11-s + (−0.926 + 0.376i)12-s + (0.352 + 1.31i)13-s + (0.237 − 0.426i)14-s + (−0.128 + 0.279i)15-s + (−0.551 − 0.833i)16-s − 1.35i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.999 - 0.0303i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.999 - 0.0303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08145 + 0.0164097i\)
\(L(\frac12)\) \(\approx\) \(1.08145 + 0.0164097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.725 + 1.21i)T \)
3 \( 1 + (-1.33 - 1.10i)T \)
good5 \( 1 + (-0.178 - 0.664i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-0.645 - 1.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.860 + 3.21i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.27 - 4.74i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 5.58iT - 17T^{2} \)
19 \( 1 + (2.49 + 2.49i)T + 19iT^{2} \)
23 \( 1 + (2.36 + 1.36i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.792 - 2.95i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.28 + 3.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.507 - 0.507i)T + 37iT^{2} \)
41 \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.949 + 0.254i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (6.13 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.601 + 0.601i)T - 53iT^{2} \)
59 \( 1 + (-4.77 + 1.28i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-10.8 - 2.90i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.110 + 0.0295i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.0447iT - 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 + (2.50 - 1.44i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.79 - 1.01i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 12.7T + 89T^{2} \)
97 \( 1 + (-4.41 - 7.63i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.22511726496190578174681556519, −11.67836927559476896307277489634, −11.11842268466073434344377625442, −9.948745517557899628793140591538, −8.978375295359900197970953449203, −8.439629627729075501834017735775, −6.93296849448018790219556671097, −4.84642622806574602812239246553, −3.54722141735992716212121319595, −2.26421437229505925502837547619, 1.57216134088585127247307718084, 3.99684407966703258503747492332, 5.68377746207426992551007159488, 6.91279575032398537831706721618, 7.907196114385051399212411506249, 8.588802707436251131109343534501, 9.777486694768657037142700186392, 10.70999011099696072180628457925, 12.56841468158307675089111558497, 13.12661028757391952998778980750

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