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Properties

Label 2-12e2-144.59-c1-0-21
Degree $2$
Conductor $144$
Sign $-0.993 + 0.112i$
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  + (−0.0112 − 1.41i)2-s + (−0.692 − 1.58i)3-s + (−1.99 + 0.0318i)4-s + (−0.759 − 2.83i)5-s + (−2.23 + 0.997i)6-s + (1.41 + 2.45i)7-s + (0.0675 + 2.82i)8-s + (−2.04 + 2.19i)9-s + (−3.99 + 1.10i)10-s + (0.212 − 0.794i)11-s + (1.43 + 3.15i)12-s + (−0.864 − 3.22i)13-s + (3.45 − 2.03i)14-s + (−3.97 + 3.16i)15-s + (3.99 − 0.127i)16-s − 7.28i·17-s + ⋯
L(s)  = 1  + (−0.00796 − 0.999i)2-s + (−0.399 − 0.916i)3-s + (−0.999 + 0.0159i)4-s + (−0.339 − 1.26i)5-s + (−0.913 + 0.407i)6-s + (0.535 + 0.927i)7-s + (0.0238 + 0.999i)8-s + (−0.680 + 0.732i)9-s + (−1.26 + 0.349i)10-s + (0.0641 − 0.239i)11-s + (0.414 + 0.910i)12-s + (−0.239 − 0.894i)13-s + (0.923 − 0.543i)14-s + (−1.02 + 0.817i)15-s + (0.999 − 0.0318i)16-s − 1.76i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.993 + 0.112i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0449287 - 0.797310i\)
\(L(\frac12)\) \(\approx\) \(0.0449287 - 0.797310i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0112 + 1.41i)T \)
3 \( 1 + (0.692 + 1.58i)T \)
good5 \( 1 + (0.759 + 2.83i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.41 - 2.45i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.212 + 0.794i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.864 + 3.22i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 7.28iT - 17T^{2} \)
19 \( 1 + (0.951 + 0.951i)T + 19iT^{2} \)
23 \( 1 + (-5.13 - 2.96i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.473 - 1.76i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.05 + 1.18i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.03 - 6.03i)T + 37iT^{2} \)
41 \( 1 + (-4.60 + 7.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.50 + 0.402i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-2.85 - 4.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.50 - 4.50i)T - 53iT^{2} \)
59 \( 1 + (10.6 - 2.85i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-8.08 - 2.16i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-11.1 + 2.99i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 2.98iT - 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + (-8.94 + 5.16i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.74 + 1.00i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 9.25T + 89T^{2} \)
97 \( 1 + (-0.148 - 0.257i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.51331504883348514823381636826, −11.74886703090803703002076940339, −11.07267239971926293273125271103, −9.348140426276109630496959847522, −8.582405151900837704803837802161, −7.60358942867547293260620127178, −5.48539074336762091806581464160, −4.89237773162699192418963230263, −2.67103122377700887137552936020, −0.921576964271878610014659149399, 3.72455595073623899511571988077, 4.56981411354965970284963886538, 6.17833966665979486087585904373, 7.02840646850534476868887238224, 8.186702619971271038328611398978, 9.527863213177132251319369601467, 10.59019786969352531752587212721, 11.14279404353452519490375424656, 12.70381958797964419264991338936, 14.18648208882719418742290050257

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