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Properties

Label 2-12e2-9.7-c5-0-17
Degree $2$
Conductor $144$
Sign $0.138 + 0.990i$
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  + (2.43 + 15.3i)3-s + (−20.8 − 36.0i)5-s + (−101. + 176. i)7-s + (−231. + 74.9i)9-s + (235. − 407. i)11-s + (−241. − 418. i)13-s + (504. − 407. i)15-s + 1.25e3·17-s − 1.97e3·19-s + (−2.95e3 − 1.13e3i)21-s + (239. + 414. i)23-s + (697. − 1.20e3i)25-s + (−1.71e3 − 3.37e3i)27-s + (580. − 1.00e3i)29-s + (−1.18e3 − 2.05e3i)31-s + ⋯
L(s)  = 1  + (0.156 + 0.987i)3-s + (−0.372 − 0.644i)5-s + (−0.784 + 1.35i)7-s + (−0.951 + 0.308i)9-s + (0.585 − 1.01i)11-s + (−0.396 − 0.686i)13-s + (0.578 − 0.468i)15-s + 1.05·17-s − 1.25·19-s + (−1.46 − 0.562i)21-s + (0.0942 + 0.163i)23-s + (0.223 − 0.386i)25-s + (−0.453 − 0.891i)27-s + (0.128 − 0.221i)29-s + (−0.221 − 0.384i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(23.0952\)
Root analytic conductor: \(4.80575\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :5/2),\ 0.138 + 0.990i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7051412224\)
\(L(\frac12)\) \(\approx\) \(0.7051412224\)
\(L(\frac{7}{2})\) 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 \( 1 + (-2.43 - 15.3i)T \)
good5 \( 1 + (20.8 + 36.0i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 + (101. - 176. i)T + (-8.40e3 - 1.45e4i)T^{2} \)
11 \( 1 + (-235. + 407. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + (241. + 418. i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 - 1.25e3T + 1.41e6T^{2} \)
19 \( 1 + 1.97e3T + 2.47e6T^{2} \)
23 \( 1 + (-239. - 414. i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (-580. + 1.00e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + (1.18e3 + 2.05e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 - 8.18e3T + 6.93e7T^{2} \)
41 \( 1 + (8.75e3 + 1.51e4i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (-1.14e4 + 1.98e4i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (8.68e3 - 1.50e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + 5.39e3T + 4.18e8T^{2} \)
59 \( 1 + (2.22e4 + 3.85e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.08e3 - 3.60e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.22e3 - 2.12e3i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 2.18e3T + 1.80e9T^{2} \)
73 \( 1 - 3.03e3T + 2.07e9T^{2} \)
79 \( 1 + (2.52e4 - 4.37e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + (2.59e4 - 4.48e4i)T + (-1.96e9 - 3.41e9i)T^{2} \)
89 \( 1 + 2.01e4T + 5.58e9T^{2} \)
97 \( 1 + (4.02e4 - 6.96e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−12.05251922109090526202342880931, −10.88885963060821254259385001569, −9.714013302972276668721867548879, −8.870792130622905506110859152398, −8.181365034657092302675821585822, −6.11997271751611225494984494789, −5.30483921103093915174228354915, −3.84905800508600194596065179138, −2.70492251476650768541823959328, −0.24417488962498909315068734932, 1.36613178061208366010859013722, 3.04073533873414901791640238208, 4.30370159884205770395918774446, 6.43273009944340340753241261466, 7.01751559226736992405711974427, 7.82168998697667160017583456598, 9.379477261819936843566035179272, 10.40283272640546746543810696916, 11.52018658123292386017393756711, 12.54562180549831147485195159186

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