Properties

Degree $2$
Conductor $144$
Sign $0.945 + 0.324i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 0.505i)2-s + (1.48 − 1.33i)4-s + (2.10 + 2.10i)5-s − 4.40·7-s + (1.28 − 2.51i)8-s + (3.84 + 1.71i)10-s + (−0.215 + 0.215i)11-s + (−2.73 − 2.73i)13-s + (−5.82 + 2.22i)14-s + (0.430 − 3.97i)16-s + 2.36i·17-s + (0.758 − 0.758i)19-s + (5.94 + 0.320i)20-s + (−0.175 + 0.393i)22-s − 1.75i·23-s + ⋯
L(s)  = 1  + (0.933 − 0.357i)2-s + (0.744 − 0.667i)4-s + (0.941 + 0.941i)5-s − 1.66·7-s + (0.456 − 0.889i)8-s + (1.21 + 0.542i)10-s + (−0.0650 + 0.0650i)11-s + (−0.758 − 0.758i)13-s + (−1.55 + 0.595i)14-s + (0.107 − 0.994i)16-s + 0.573i·17-s + (0.174 − 0.174i)19-s + (1.32 + 0.0717i)20-s + (−0.0374 + 0.0839i)22-s − 0.366i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.945 + 0.324i$
Motivic weight: \(1\)
Character: $\chi_{144} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.945 + 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76228 - 0.294013i\)
\(L(\frac12)\) \(\approx\) \(1.76228 - 0.294013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.505i)T \)
3 \( 1 \)
good5 \( 1 + (-2.10 - 2.10i)T + 5iT^{2} \)
7 \( 1 + 4.40T + 7T^{2} \)
11 \( 1 + (0.215 - 0.215i)T - 11iT^{2} \)
13 \( 1 + (2.73 + 2.73i)T + 13iT^{2} \)
17 \( 1 - 2.36iT - 17T^{2} \)
19 \( 1 + (-0.758 + 0.758i)T - 19iT^{2} \)
23 \( 1 + 1.75iT - 23T^{2} \)
29 \( 1 + (5.54 - 5.54i)T - 29iT^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 + (-3.10 + 3.10i)T - 37iT^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + (3.54 + 3.54i)T + 43iT^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 + (2.71 + 2.71i)T + 53iT^{2} \)
59 \( 1 + (-3.40 + 3.40i)T - 59iT^{2} \)
61 \( 1 + (1.75 + 1.75i)T + 61iT^{2} \)
67 \( 1 + (-9.11 + 9.11i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + 0.482iT - 73T^{2} \)
79 \( 1 + 6.88iT - 79T^{2} \)
83 \( 1 + (-4.79 - 4.79i)T + 83iT^{2} \)
89 \( 1 + 7.00T + 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−12.87703332355453745395949373874, −12.53224179578012211433808275662, −10.86001923449454881357873558700, −10.20096827810122274040801516472, −9.421366104174466807929942604252, −7.15799245774610958890492758869, −6.38319020076084856130429720740, −5.42758558814453825204715243431, −3.48396164761634913952741917454, −2.52957698623826832091819826160, 2.51819270110812640098923984345, 4.15516711459903705594006560262, 5.54386122765817628547916916608, 6.34051154731749513323572370581, 7.56053618498693285674061338573, 9.273619415682010405566945382296, 9.778976656497763731254951888697, 11.51594547530035450589356959224, 12.55612919562958522833681767146, 13.21460542919134488713935173521

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