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Properties

Label 2-12e2-144.59-c1-0-9
Degree $2$
Conductor $144$
Sign $0.265 - 0.964i$
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  + (0.378 + 1.36i)2-s + (1.53 − 0.800i)3-s + (−1.71 + 1.03i)4-s + (1.00 + 3.73i)5-s + (1.67 + 1.79i)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 + (0.0566 − 0.211i)11-s + (−1.80 + 2.95i)12-s + (0.727 + 2.71i)13-s + (3.33 − 3.39i)14-s + (4.52 + 4.93i)15-s + (1.87 − 3.53i)16-s − 4.23i·17-s + ⋯
L(s)  = 1  + (0.267 + 0.963i)2-s + (0.886 − 0.461i)3-s + (−0.856 + 0.515i)4-s + (0.447 + 1.67i)5-s + (0.682 + 0.730i)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 + (0.0170 − 0.0637i)11-s + (−0.521 + 0.853i)12-s + (0.201 + 0.753i)13-s + (0.891 − 0.907i)14-s + (1.16 + 1.27i)15-s + (0.468 − 0.883i)16-s − 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.265 - 0.964i$
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: \((2,\ 144,\ (\ :1/2),\ 0.265 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18011 + 0.899050i\)
\(L(\frac12)\) \(\approx\) \(1.18011 + 0.899050i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.378 - 1.36i)T \)
3 \( 1 + (-1.53 + 0.800i)T \)
good5 \( 1 + (-1.00 - 3.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.68 + 2.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.0566 + 0.211i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.727 - 2.71i)T + (-11.2 + 6.5i)T^{2} \)
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 \( 1 + (-0.545 + 2.03i)T + (-25.1 - 14.5i)T^{2} \)
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 \( 1 + (-4.68 - 1.25i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.34 - 4.07i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.58 - 7.58i)T - 53iT^{2} \)
59 \( 1 + (5.34 - 1.43i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.69 - 2.33i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (5.17 - 1.38i)T + (58.0 - 33.5i)T^{2} \)
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 \( 1 + (-4.50 - 1.20i)T + (71.8 + 41.5i)T^{2} \)
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

−13.77368185238442103789536604657, −12.89117056433432765270443414604, −11.23788973627832050591185168178, −9.924098402317956971420181920475, −9.147065745115403877405652359831, −7.42483292976763930010175048351, −7.09488009497548372383717681404, −6.18620560630427865622664722228, −3.98037181299599489140733925503, −2.90963407765426067896929545031, 1.87304343917988474420704737661, 3.43474588909290215740537267672, 4.85239631758532937323721526038, 5.74058254143118900722036536563, 8.423039534620236892552879264030, 8.858239221893061995774813877926, 9.665698701428368223811859460219, 10.68468157813440859460110127906, 12.47943648396070201344321036281, 12.65003917416244300108784267742

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