Properties

Degree $2$
Conductor $144$
Sign $0.745 - 0.666i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.957 + 1.04i)2-s + (−0.167 − 1.99i)4-s + (0.236 − 0.236i)5-s + 3.27·7-s + (2.23 + 1.73i)8-s + (0.0197 + 0.472i)10-s + (2.58 + 2.58i)11-s + (−1.70 + 1.70i)13-s + (−3.13 + 3.41i)14-s + (−3.94 + 0.665i)16-s − 7.05i·17-s + (3.04 + 3.04i)19-s + (−0.510 − 0.431i)20-s + (−5.16 + 0.215i)22-s + 1.47i·23-s + ⋯
L(s)  = 1  + (−0.676 + 0.736i)2-s + (−0.0835 − 0.996i)4-s + (0.105 − 0.105i)5-s + 1.23·7-s + (0.790 + 0.613i)8-s + (0.00624 + 0.149i)10-s + (0.778 + 0.778i)11-s + (−0.473 + 0.473i)13-s + (−0.838 + 0.912i)14-s + (−0.986 + 0.166i)16-s − 1.71i·17-s + (0.697 + 0.697i)19-s + (−0.114 − 0.0964i)20-s + (−1.10 + 0.0460i)22-s + 0.307i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.745 - 0.666i$
Motivic weight: \(1\)
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848376 + 0.324119i\)
\(L(\frac12)\) \(\approx\) \(0.848376 + 0.324119i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.957 - 1.04i)T \)
3 \( 1 \)
good5 \( 1 + (-0.236 + 0.236i)T - 5iT^{2} \)
7 \( 1 - 3.27T + 7T^{2} \)
11 \( 1 + (-2.58 - 2.58i)T + 11iT^{2} \)
13 \( 1 + (1.70 - 1.70i)T - 13iT^{2} \)
17 \( 1 + 7.05iT - 17T^{2} \)
19 \( 1 + (-3.04 - 3.04i)T + 19iT^{2} \)
23 \( 1 - 1.47iT - 23T^{2} \)
29 \( 1 + (2.98 + 2.98i)T + 29iT^{2} \)
31 \( 1 + 8.02iT - 31T^{2} \)
37 \( 1 + (7.93 + 7.93i)T + 37iT^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 + (4.61 - 4.61i)T - 43iT^{2} \)
47 \( 1 + 7.13T + 47T^{2} \)
53 \( 1 + (5.81 - 5.81i)T - 53iT^{2} \)
59 \( 1 + (-7.46 - 7.46i)T + 59iT^{2} \)
61 \( 1 + (4.04 - 4.04i)T - 61iT^{2} \)
67 \( 1 + (2.90 + 2.90i)T + 67iT^{2} \)
71 \( 1 - 1.02iT - 71T^{2} \)
73 \( 1 + 4.08iT - 73T^{2} \)
79 \( 1 + 5.36iT - 79T^{2} \)
83 \( 1 + (3.93 - 3.93i)T - 83iT^{2} \)
89 \( 1 + 2.35T + 89T^{2} \)
97 \( 1 - 9.97T + 97T^{2} \)
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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

−13.60991336916697375824256459810, −11.87542305701360114167001028699, −11.23819679833684546955256862785, −9.734564246892325468804344070013, −9.177774931834446767647892342932, −7.75192844161350779332853086773, −7.13986512200461492705201634079, −5.54624297086259203620664138904, −4.54150835306952303932876412721, −1.73326008034501168371744012335, 1.60107677076249520353303756797, 3.42577724117731872585422721826, 4.94052685595253815119049126499, 6.73181547449944922272193093975, 8.158000561862939549604258621448, 8.681425322338248942052411601031, 10.10698114669409711629479069664, 10.93911078291367855116840750000, 11.76225391784834443935327308666, 12.70578779572280557897486233550

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