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Properties

Label 2-12e2-9.7-c3-0-3
Degree $2$
Conductor $144$
Sign $0.173 - 0.984i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.19i·3-s + (4.5 + 7.79i)5-s + (−15.5 + 26.8i)7-s − 27·9-s + (−7.5 + 12.9i)11-s + (18.5 + 32.0i)13-s + (40.5 − 23.3i)15-s − 42·17-s + 28·19-s + (139.5 + 80.5i)21-s + (97.5 + 168. i)23-s + (22 − 38.1i)25-s + 140. i·27-s + (−55.5 + 96.1i)29-s + (−102.5 − 177. i)31-s + ⋯
L(s)  = 1  − 0.999i·3-s + (0.402 + 0.697i)5-s + (−0.836 + 1.44i)7-s − 9-s + (−0.205 + 0.356i)11-s + (0.394 + 0.683i)13-s + (0.697 − 0.402i)15-s − 0.599·17-s + 0.338·19-s + (1.44 + 0.836i)21-s + (0.883 + 1.53i)23-s + (0.175 − 0.304i)25-s + 1.00i·27-s + (−0.355 + 0.615i)29-s + (−0.593 − 1.02i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.896013 + 0.751844i\)
\(L(\frac12)\) \(\approx\) \(0.896013 + 0.751844i\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.19iT \)
good5 \( 1 + (-4.5 - 7.79i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (15.5 - 26.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (7.5 - 12.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-18.5 - 32.0i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 42T + 4.91e3T^{2} \)
19 \( 1 - 28T + 6.85e3T^{2} \)
23 \( 1 + (-97.5 - 168. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (55.5 - 96.1i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (102.5 + 177. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 166T + 5.06e4T^{2} \)
41 \( 1 + (-130.5 - 226. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (21.5 - 37.2i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (-88.5 + 153. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 114T + 1.48e5T^{2} \)
59 \( 1 + (-79.5 - 137. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (95.5 - 165. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (210.5 + 364. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 156T + 3.57e5T^{2} \)
73 \( 1 - 182T + 3.89e5T^{2} \)
79 \( 1 + (-566.5 + 981. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (541.5 - 937. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + (-450.5 + 780. i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.90701471902972991806865397023, −11.93359629859719486312321824100, −11.08685161137955572248072520465, −9.535408448848577925756927779509, −8.792069137818794532315345192120, −7.29865575009638333516137325121, −6.39345413794908118157227988492, −5.51497934120753494774056330432, −3.09811646190709306470295566380, −1.99972156492406522762971089083, 0.56050723159954526261225446280, 3.20984528326290633803218854987, 4.37782043660692309743559654858, 5.57359751176674089642378312113, 6.93840614120262923114859104892, 8.483692024779758458543731784478, 9.387684353934934106698772709325, 10.44434004476588536725363606051, 10.93515162410398207770947964488, 12.63454411533842024513409871731

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