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Properties

Label 2-12e2-36.31-c2-0-3
Degree $2$
Conductor $144$
Sign $-0.506 - 0.862i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.531 + 2.95i)3-s + (−3.31 + 5.73i)5-s + (8.46 − 4.88i)7-s + (−8.43 + 3.13i)9-s + (−10.0 + 5.80i)11-s + (−8.93 + 15.4i)13-s + (−18.6 − 6.72i)15-s + 2.37·17-s − 14.0i·19-s + (18.9 + 22.4i)21-s + (35.9 + 20.7i)23-s + (−9.43 − 16.3i)25-s + (−13.7 − 23.2i)27-s + (−5.68 − 9.85i)29-s + (18.0 + 10.4i)31-s + ⋯
L(s)  = 1  + (0.177 + 0.984i)3-s + (−0.662 + 1.14i)5-s + (1.20 − 0.698i)7-s + (−0.937 + 0.348i)9-s + (−0.914 + 0.528i)11-s + (−0.687 + 1.19i)13-s + (−1.24 − 0.448i)15-s + 0.139·17-s − 0.738i·19-s + (0.901 + 1.06i)21-s + (1.56 + 0.902i)23-s + (−0.377 − 0.653i)25-s + (−0.509 − 0.860i)27-s + (−0.196 − 0.339i)29-s + (0.581 + 0.335i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.635185 + 1.10922i\)
\(L(\frac12)\) \(\approx\) \(0.635185 + 1.10922i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.531 - 2.95i)T \)
good5 \( 1 + (3.31 - 5.73i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-8.46 + 4.88i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (10.0 - 5.80i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (8.93 - 15.4i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 2.37T + 289T^{2} \)
19 \( 1 + 14.0iT - 361T^{2} \)
23 \( 1 + (-35.9 - 20.7i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (5.68 + 9.85i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-18.0 - 10.4i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 35.7T + 1.36e3T^{2} \)
41 \( 1 + (2.62 - 4.55i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-54.4 + 31.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (4.28 - 2.47i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 75.7T + 2.80e3T^{2} \)
59 \( 1 + (50.3 + 29.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-5.93 - 10.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-20.6 - 11.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 46.4iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + (-18.0 + 10.4i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-24.4 + 14.0i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 73.0T + 7.92e3T^{2} \)
97 \( 1 + (-71.1 - 123. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.51137991080381712371715683248, −11.69166245306548426576825656128, −11.08870008708350700484134228023, −10.37497405013279233478892792787, −9.183321052090391091510485139821, −7.76880026992806303957827454313, −7.09065330023951958346720692727, −5.06093931809004450284251465967, −4.16746836574897284952676142850, −2.66401950240505309740511491590, 0.857134315986928163426970706320, 2.68977034013878890351765128623, 4.85058665254368919631829495569, 5.70270498028242895202895573746, 7.64813743845697050971023141631, 8.149028348196760680229233769575, 8.922671704617376534014139704951, 10.75715879043667583705057275817, 11.81571034938120122496911460536, 12.56253143153233568403577020297

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