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Properties

Label 2-12e2-144.83-c1-0-20
Degree $2$
Conductor $144$
Sign $0.386 + 0.922i$
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  + (1.20 − 0.742i)2-s + (0.819 − 1.52i)3-s + (0.896 − 1.78i)4-s + (−0.206 + 0.769i)5-s + (−0.146 − 2.44i)6-s + (−2.17 + 3.76i)7-s + (−0.248 − 2.81i)8-s + (−1.65 − 2.50i)9-s + (0.323 + 1.07i)10-s + (1.05 + 3.93i)11-s + (−1.99 − 2.83i)12-s + (−0.454 + 1.69i)13-s + (0.180 + 6.15i)14-s + (1.00 + 0.945i)15-s + (−2.39 − 3.20i)16-s − 6.68i·17-s + ⋯
L(s)  = 1  + (0.850 − 0.525i)2-s + (0.473 − 0.880i)3-s + (0.448 − 0.893i)4-s + (−0.0922 + 0.344i)5-s + (−0.0598 − 0.998i)6-s + (−0.822 + 1.42i)7-s + (−0.0879 − 0.996i)8-s + (−0.551 − 0.833i)9-s + (0.102 + 0.341i)10-s + (0.318 + 1.18i)11-s + (−0.575 − 0.818i)12-s + (−0.126 + 0.470i)13-s + (0.0482 + 1.64i)14-s + (0.259 + 0.244i)15-s + (−0.598 − 0.801i)16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47895 - 0.983969i\)
\(L(\frac12)\) \(\approx\) \(1.47895 - 0.983969i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.742i)T \)
3 \( 1 + (-0.819 + 1.52i)T \)
good5 \( 1 + (0.206 - 0.769i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (2.17 - 3.76i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.05 - 3.93i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.454 - 1.69i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 6.68iT - 17T^{2} \)
19 \( 1 + (0.708 - 0.708i)T - 19iT^{2} \)
23 \( 1 + (-3.88 + 2.24i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.06 + 3.98i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.94 - 2.85i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.51 - 1.51i)T - 37iT^{2} \)
41 \( 1 + (-1.36 - 2.36i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.60 - 2.30i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.23 + 2.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.68 - 1.68i)T + 53iT^{2} \)
59 \( 1 + (1.00 + 0.269i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.97 + 0.528i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.01 - 2.14i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 8.05iT - 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 + (-11.9 - 6.91i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.05 + 0.817i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.71T + 89T^{2} \)
97 \( 1 + (-3.66 + 6.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.73582303675805961308406587382, −12.18900673071176270854364804637, −11.39762999894970801446289818461, −9.689017004848581295458202100068, −9.044161028101068368240311520837, −7.14310856036029314145022925849, −6.47446581729056576556082349034, −5.07324510598912633385392258934, −3.17211626326148265993543794902, −2.19106341210851240441535424720, 3.35527998323633432085115989534, 3.99504771358009772394264269371, 5.41822489432710945701831713207, 6.72084212213591984432471620635, 8.012903923415235566376048363644, 8.958717740724102404955489004210, 10.44618986301571955520028084572, 11.09380932163319221974365747341, 12.77366087607845957305971112630, 13.37838458945844129937571863927

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