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Properties

Label 2-12e2-9.4-c5-0-0
Degree $2$
Conductor $144$
Sign $-0.915 + 0.403i$
Analytic cond. $23.0952$
Root an. cond. $4.80575$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (12.2 + 9.62i)3-s + (−14.0 + 24.3i)5-s + (−75.7 − 131. i)7-s + (57.8 + 236. i)9-s + (−138. − 240. i)11-s + (−291. + 505. i)13-s + (−407. + 163. i)15-s − 1.61e3·17-s − 1.36e3·19-s + (333. − 2.33e3i)21-s + (428. − 741. i)23-s + (1.16e3 + 2.02e3i)25-s + (−1.56e3 + 3.45e3i)27-s + (−4.26e3 − 7.39e3i)29-s + (1.46e3 − 2.54e3i)31-s + ⋯
L(s)  = 1  + (0.786 + 0.617i)3-s + (−0.251 + 0.435i)5-s + (−0.583 − 1.01i)7-s + (0.238 + 0.971i)9-s + (−0.346 − 0.599i)11-s + (−0.479 + 0.829i)13-s + (−0.467 + 0.187i)15-s − 1.35·17-s − 0.869·19-s + (0.164 − 1.15i)21-s + (0.168 − 0.292i)23-s + (0.373 + 0.646i)25-s + (−0.412 + 0.911i)27-s + (−0.942 − 1.63i)29-s + (0.274 − 0.475i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.915 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.915 + 0.403i$
Analytic conductor: \(23.0952\)
Root analytic conductor: \(4.80575\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :5/2),\ -0.915 + 0.403i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1147941118\)
\(L(\frac12)\) \(\approx\) \(0.1147941118\)
\(L(\frac{7}{2})\) 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 \( 1 + (-12.2 - 9.62i)T \)
good5 \( 1 + (14.0 - 24.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + (75.7 + 131. i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (138. + 240. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + (291. - 505. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + 1.61e3T + 1.41e6T^{2} \)
19 \( 1 + 1.36e3T + 2.47e6T^{2} \)
23 \( 1 + (-428. + 741. i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (4.26e3 + 7.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 + (-1.46e3 + 2.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 - 4.03e3T + 6.93e7T^{2} \)
41 \( 1 + (9.44e3 - 1.63e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (1.01e4 + 1.75e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (147. + 256. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 - 3.03e3T + 4.18e8T^{2} \)
59 \( 1 + (8.61e3 - 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.28e4 + 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.31e4 - 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 7.66e4T + 1.80e9T^{2} \)
73 \( 1 - 1.49e3T + 2.07e9T^{2} \)
79 \( 1 + (-4.96e4 - 8.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + (-2.50e4 - 4.33e4i)T + (-1.96e9 + 3.41e9i)T^{2} \)
89 \( 1 - 1.36e5T + 5.58e9T^{2} \)
97 \( 1 + (-3.33e4 - 5.77e4i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−13.27072587384178872218512473643, −11.44126452536823053888049910487, −10.62913650339895169624428528382, −9.715008071643110245197055012458, −8.677528521188654923537407872996, −7.50713396665185349486389559342, −6.49540494741178813874726133330, −4.57466718075238047062949829454, −3.66629215002745305837316053926, −2.33155470659926820900888676885, 0.03213018443031007644988130060, 1.99042911986332009308469380558, 3.11070876428603001369726244492, 4.79027629950004561515126948848, 6.28614866194015164732947775200, 7.39778275315913151030373350427, 8.598456153068047138886846087382, 9.164868867278498056566564778830, 10.47831406511256386324764681455, 12.03269863177538258136618885918

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