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Properties

Label 2-12e2-144.5-c2-0-0
Degree $2$
Conductor $144$
Sign $-0.518 + 0.855i$
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  + (−1.26 + 1.54i)2-s + (−2.98 − 0.310i)3-s + (−0.803 − 3.91i)4-s + (2.23 + 8.33i)5-s + (4.25 − 4.23i)6-s + (−3.69 + 2.13i)7-s + (7.08 + 3.70i)8-s + (8.80 + 1.85i)9-s + (−15.7 − 7.07i)10-s + (−15.6 − 4.18i)11-s + (1.17 + 11.9i)12-s + (−5.18 − 19.3i)13-s + (1.36 − 8.42i)14-s + (−4.07 − 25.5i)15-s + (−14.7 + 6.29i)16-s − 10.6i·17-s + ⋯
L(s)  = 1  + (−0.632 + 0.774i)2-s + (−0.994 − 0.103i)3-s + (−0.200 − 0.979i)4-s + (0.446 + 1.66i)5-s + (0.709 − 0.705i)6-s + (−0.528 + 0.304i)7-s + (0.885 + 0.463i)8-s + (0.978 + 0.206i)9-s + (−1.57 − 0.707i)10-s + (−1.42 − 0.380i)11-s + (0.0982 + 0.995i)12-s + (−0.398 − 1.48i)13-s + (0.0975 − 0.601i)14-s + (−0.271 − 1.70i)15-s + (−0.919 + 0.393i)16-s − 0.627i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.518 + 0.855i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.518 + 0.855i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.518 + 0.855i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0483554 - 0.0858669i\)
\(L(\frac12)\) \(\approx\) \(0.0483554 - 0.0858669i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 1.54i)T \)
3 \( 1 + (2.98 + 0.310i)T \)
good5 \( 1 + (-2.23 - 8.33i)T + (-21.6 + 12.5i)T^{2} \)
7 \( 1 + (3.69 - 2.13i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (15.6 + 4.18i)T + (104. + 60.5i)T^{2} \)
13 \( 1 + (5.18 + 19.3i)T + (-146. + 84.5i)T^{2} \)
17 \( 1 + 10.6iT - 289T^{2} \)
19 \( 1 + (12.2 - 12.2i)T - 361iT^{2} \)
23 \( 1 + (-9.25 + 16.0i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (6.80 - 25.4i)T + (-728. - 420.5i)T^{2} \)
31 \( 1 + (-6.33 + 10.9i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (18.0 + 18.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (7.43 - 12.8i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-4.70 + 17.5i)T + (-1.60e3 - 924.5i)T^{2} \)
47 \( 1 + (16.1 - 9.32i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (38.0 - 38.0i)T - 2.80e3iT^{2} \)
59 \( 1 + (-16.9 - 63.4i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (11.9 + 3.19i)T + (3.22e3 + 1.86e3i)T^{2} \)
67 \( 1 + (-8.10 - 30.2i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 38.1T + 5.04e3T^{2} \)
73 \( 1 + 127. iT - 5.32e3T^{2} \)
79 \( 1 + (-7.54 - 13.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (29.3 - 109. i)T + (-5.96e3 - 3.44e3i)T^{2} \)
89 \( 1 + 102.T + 7.92e3T^{2} \)
97 \( 1 + (-40.9 - 70.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−13.60600801221855166962696569816, −12.60698069773996994207775723482, −10.82636074381040360711797463101, −10.60827180405020749124736991399, −9.751198107018863555573709012072, −7.924436805519212466142198845370, −7.03049082967990827738590213233, −6.06226420982059796599049294088, −5.30980680235833741090269793430, −2.72349067259384619326845852285, 0.085329577450853863997556623340, 1.78272093718974027419767074435, 4.29059402359685759604347820408, 5.15500511668264632243843822891, 6.82385489315870450421013268340, 8.234394849142078400952970551451, 9.431496379003762309953800748097, 10.00330635757638206753536473736, 11.19216075964984407266384551185, 12.20456422856955450161158916456

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