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Properties

Label 2-12e2-144.85-c1-0-2
Degree $2$
Conductor $144$
Sign $-0.926 - 0.377i$
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 + 0.736i)2-s + (1.05 + 1.37i)3-s + (0.916 − 1.77i)4-s + (−4.10 + 1.10i)5-s + (−2.28 − 0.885i)6-s + (−1.63 + 0.942i)7-s + (0.201 + 2.82i)8-s + (−0.782 + 2.89i)9-s + (4.15 − 4.35i)10-s + (0.317 − 1.18i)11-s + (3.40 − 0.611i)12-s + (0.620 + 2.31i)13-s + (1.27 − 2.34i)14-s + (−5.84 − 4.49i)15-s + (−2.32 − 3.25i)16-s + 3.44·17-s + ⋯
L(s)  = 1  + (−0.853 + 0.520i)2-s + (0.607 + 0.794i)3-s + (0.458 − 0.888i)4-s + (−1.83 + 0.492i)5-s + (−0.932 − 0.361i)6-s + (−0.617 + 0.356i)7-s + (0.0713 + 0.997i)8-s + (−0.260 + 0.965i)9-s + (1.31 − 1.37i)10-s + (0.0956 − 0.356i)11-s + (0.984 − 0.176i)12-s + (0.171 + 0.641i)13-s + (0.341 − 0.625i)14-s + (−1.50 − 1.15i)15-s + (−0.580 − 0.814i)16-s + 0.835·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.926 - 0.377i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.926 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100761 + 0.514206i\)
\(L(\frac12)\) \(\approx\) \(0.100761 + 0.514206i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 - 0.736i)T \)
3 \( 1 + (-1.05 - 1.37i)T \)
good5 \( 1 + (4.10 - 1.10i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.63 - 0.942i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.317 + 1.18i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.620 - 2.31i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 3.44T + 17T^{2} \)
19 \( 1 + (4.17 - 4.17i)T - 19iT^{2} \)
23 \( 1 + (-1.34 - 0.778i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.28 - 0.343i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.25 + 2.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.77 - 5.77i)T + 37iT^{2} \)
41 \( 1 + (-3.43 - 1.98i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.793 + 2.96i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-0.230 - 0.399i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.61 - 5.61i)T + 53iT^{2} \)
59 \( 1 + (10.9 - 2.92i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.74 + 1.27i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.0767 + 0.286i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 - 0.0279iT - 73T^{2} \)
79 \( 1 + (-2.19 - 3.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.41 - 0.915i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 4.74iT - 89T^{2} \)
97 \( 1 + (-4.17 - 7.22i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.03415459746289968785397157753, −12.21063169649061499601784193007, −11.22056331054082628796263121614, −10.40242995614552471666311878810, −9.235692481697062445595258891762, −8.272145903301654786738999516493, −7.58749737909118975612921985278, −6.21820085873303460697637077625, −4.36576302659338223281877271922, −3.08626205876825259859550779783, 0.64962139613680395618178861493, 3.02467508930523035143136874885, 4.09622919034647255069324727205, 6.80081808627792922117509786689, 7.64213728178773947328595635317, 8.351914001883291568506129755252, 9.291114550620784427854965073650, 10.73399616051954191631680133346, 11.76613858522016424573750353780, 12.58182770051984052632245380306

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