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Properties

Label 2-12e2-144.133-c1-0-20
Degree $2$
Conductor $144$
Sign $-0.417 + 0.908i$
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  + (0.282 − 1.38i)2-s + (0.944 − 1.45i)3-s + (−1.84 − 0.783i)4-s + (0.131 − 0.491i)5-s + (−1.74 − 1.71i)6-s + (2.40 + 1.38i)7-s + (−1.60 + 2.32i)8-s + (−1.21 − 2.74i)9-s + (−0.644 − 0.321i)10-s + (−3.96 + 1.06i)11-s + (−2.87 + 1.93i)12-s + (2.22 + 0.596i)13-s + (2.60 − 2.94i)14-s + (−0.589 − 0.655i)15-s + (2.77 + 2.88i)16-s + 2.87·17-s + ⋯
L(s)  = 1  + (0.199 − 0.979i)2-s + (0.545 − 0.838i)3-s + (−0.920 − 0.391i)4-s + (0.0589 − 0.219i)5-s + (−0.712 − 0.701i)6-s + (0.909 + 0.524i)7-s + (−0.567 + 0.823i)8-s + (−0.405 − 0.913i)9-s + (−0.203 − 0.101i)10-s + (−1.19 + 0.320i)11-s + (−0.829 + 0.557i)12-s + (0.617 + 0.165i)13-s + (0.695 − 0.785i)14-s + (−0.152 − 0.169i)15-s + (0.693 + 0.720i)16-s + 0.698·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.417 + 0.908i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720046 - 1.12338i\)
\(L(\frac12)\) \(\approx\) \(0.720046 - 1.12338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.282 + 1.38i)T \)
3 \( 1 + (-0.944 + 1.45i)T \)
good5 \( 1 + (-0.131 + 0.491i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.40 - 1.38i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.96 - 1.06i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.22 - 0.596i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 2.87T + 17T^{2} \)
19 \( 1 + (3.48 - 3.48i)T - 19iT^{2} \)
23 \( 1 + (-3.85 + 2.22i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.57 + 5.88i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (1.28 + 2.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.64 - 7.64i)T + 37iT^{2} \)
41 \( 1 + (4.84 - 2.79i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.40 - 0.911i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (4.94 - 8.56i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.86 + 2.86i)T + 53iT^{2} \)
59 \( 1 + (0.577 - 2.15i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.28 + 4.79i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (14.7 + 3.96i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 + 11.3iT - 73T^{2} \)
79 \( 1 + (1.56 - 2.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.95 + 11.0i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 - 2.37iT - 89T^{2} \)
97 \( 1 + (5.04 - 8.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.87182097048415884552856167451, −11.89203112476470936839699102968, −10.98687377346356108728709427579, −9.730708575953472786446199929012, −8.507622746978573703950137562004, −7.920844517617617079812759589049, −6.00905055796535592170310946207, −4.72807734617566842166023883886, −2.95818968819334706903213127332, −1.63219223543811628058494970752, 3.21344959276496611471369562690, 4.62430597722281660017986436298, 5.50215674100753524612431628114, 7.21908028773314363000473199941, 8.184615954157295932217410125400, 8.955128455076791721042304359424, 10.36940999215746982747064932239, 11.05450998438198275376105307855, 12.91295963773641956214845641294, 13.69228063137203936045192111726

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