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Properties

Label 2-12e2-144.83-c1-0-18
Degree $2$
Conductor $144$
Sign $0.770 + 0.637i$
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  + (1.41 + 0.0121i)2-s + (−1.32 − 1.11i)3-s + (1.99 + 0.0343i)4-s + (0.323 − 1.20i)5-s + (−1.86 − 1.59i)6-s + (0.140 − 0.242i)7-s + (2.82 + 0.0728i)8-s + (0.513 + 2.95i)9-s + (0.471 − 1.70i)10-s + (−0.823 − 3.07i)11-s + (−2.61 − 2.27i)12-s + (−0.740 + 2.76i)13-s + (0.201 − 0.341i)14-s + (−1.77 + 1.23i)15-s + (3.99 + 0.137i)16-s + 3.72i·17-s + ⋯
L(s)  = 1  + (0.999 + 0.00858i)2-s + (−0.765 − 0.643i)3-s + (0.999 + 0.0171i)4-s + (0.144 − 0.539i)5-s + (−0.759 − 0.650i)6-s + (0.0530 − 0.0918i)7-s + (0.999 + 0.0257i)8-s + (0.171 + 0.985i)9-s + (0.149 − 0.538i)10-s + (−0.248 − 0.926i)11-s + (−0.754 − 0.656i)12-s + (−0.205 + 0.766i)13-s + (0.0538 − 0.0913i)14-s + (−0.457 + 0.319i)15-s + (0.999 + 0.0343i)16-s + 0.902i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.770 + 0.637i$
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: \((2,\ 144,\ (\ :1/2),\ 0.770 + 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49513 - 0.538354i\)
\(L(\frac12)\) \(\approx\) \(1.49513 - 0.538354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0121i)T \)
3 \( 1 + (1.32 + 1.11i)T \)
good5 \( 1 + (-0.323 + 1.20i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.140 + 0.242i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.823 + 3.07i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.740 - 2.76i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 - 3.72iT - 17T^{2} \)
19 \( 1 + (4.10 - 4.10i)T - 19iT^{2} \)
23 \( 1 + (1.57 - 0.909i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.02 + 3.83i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (8.81 - 5.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.76 + 1.76i)T - 37iT^{2} \)
41 \( 1 + (2.66 + 4.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.89 + 1.84i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.48 + 9.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.58 + 8.58i)T + 53iT^{2} \)
59 \( 1 + (-5.38 - 1.44i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.23 - 1.66i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-3.75 - 1.00i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.6iT - 71T^{2} \)
73 \( 1 + 9.30iT - 73T^{2} \)
79 \( 1 + (8.70 + 5.02i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.19 + 0.588i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + (-9.19 + 15.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.85666827022406976180954913211, −12.28218556699100926308258039282, −11.19634059044950356897751882868, −10.45184659936659459116923699321, −8.563575656572489335368907451707, −7.34321387756850576491301999874, −6.16200950980112530388631586567, −5.38153419997415473447140889928, −3.99577005890824845678420631962, −1.84782625110706627709846080669, 2.73281295013410113144649432848, 4.34214659874672756522016245418, 5.30098872419721911101482346049, 6.47075002652503423601289688082, 7.45649423207058823323166271221, 9.440129452135839661524377013618, 10.58160440087764998772251965816, 11.13032382503168795532345764805, 12.32741790940360711356934793732, 12.98830233070827418788979234774

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