Properties

Label 2-12e2-9.4-c1-0-3
Degree $2$
Conductor $144$
Sign $0.939 + 0.342i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 0.866i)3-s + (1 + 1.73i)7-s + (1.5 − 2.59i)9-s + (−1.5 − 2.59i)11-s + (−1 + 1.73i)13-s − 3·17-s + 19-s + (3 + 1.73i)21-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 5.19i·27-s + (−3 − 5.19i)29-s + (−2 + 3.46i)31-s + (−4.5 − 2.59i)33-s − 4·37-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.377 + 0.654i)7-s + (0.5 − 0.866i)9-s + (−0.452 − 0.783i)11-s + (−0.277 + 0.480i)13-s − 0.727·17-s + 0.229·19-s + (0.654 + 0.377i)21-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−0.557 − 0.964i)29-s + (−0.359 + 0.622i)31-s + (−0.783 − 0.452i)33-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37906 - 0.243166i\)
\(L(\frac12)\) \(\approx\) \(1.37906 - 0.243166i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.5 + 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.26753273033277262417649451966, −12.08741449551169059831330125042, −11.21134912092071582860274237943, −9.699742114448970930952488412869, −8.773992186911460946511155514314, −7.914504181237864992656505890458, −6.72808141967301058242909305741, −5.31075452569347015793858506264, −3.53519034210908129381929486333, −2.04901078629379074055897362975, 2.35108135638940597967821092355, 4.00947033884517203424427915554, 5.05711936670991334027722453981, 7.00489071735715296868076371624, 7.957456630593232112232971517021, 8.985259579193792400133529119334, 10.20755662618736127992191657134, 10.74238199510750267337032779634, 12.32959609175909797451294911946, 13.29412028177970020405968736759

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