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Properties

Label 2-12e2-144.61-c1-0-17
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: $\chi_{144} (61, \cdot )$
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

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

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