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Properties

Label 2-12e2-9.7-c1-0-3
Degree $2$
Conductor $144$
Sign $0.283 + 0.959i$
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  + (−1.68 + 0.396i)3-s + (−1.18 − 2.05i)5-s + (2.18 − 3.78i)7-s + (2.68 − 1.33i)9-s + (0.5 − 0.866i)11-s + (0.186 + 0.322i)13-s + (2.81 + 2.99i)15-s − 5.37·17-s − 0.627·19-s + (−2.18 + 7.25i)21-s + (−0.186 − 0.322i)23-s + (−0.313 + 0.543i)25-s + (−4 + 3.31i)27-s + (2.18 − 3.78i)29-s + (3.18 + 5.51i)31-s + ⋯
L(s)  = 1  + (−0.973 + 0.228i)3-s + (−0.530 − 0.918i)5-s + (0.826 − 1.43i)7-s + (0.895 − 0.445i)9-s + (0.150 − 0.261i)11-s + (0.0516 + 0.0894i)13-s + (0.726 + 0.773i)15-s − 1.30·17-s − 0.144·19-s + (−0.477 + 1.58i)21-s + (−0.0388 − 0.0672i)23-s + (−0.0627 + 0.108i)25-s + (−0.769 + 0.638i)27-s + (0.405 − 0.703i)29-s + (0.572 + 0.991i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.283 + 0.959i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.283 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.618997 - 0.462709i\)
\(L(\frac12)\) \(\approx\) \(0.618997 - 0.462709i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 - 0.396i)T \)
good5 \( 1 + (1.18 + 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.18 + 3.78i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.186 - 0.322i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.37T + 17T^{2} \)
19 \( 1 + 0.627T + 19T^{2} \)
23 \( 1 + (0.186 + 0.322i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.18 + 3.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.18 - 5.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.74T + 37T^{2} \)
41 \( 1 + (-5.87 - 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.872 - 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.18 + 3.78i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.744T + 53T^{2} \)
59 \( 1 + (-3.5 - 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.18 - 2.05i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.87 + 3.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (3.18 - 5.51i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.81 + 8.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (0.872 - 1.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.83863838795933213855587608934, −11.66564627597718803806258169392, −11.05754279120628328255934552761, −10.06559759566031401195821034335, −8.653446361384281559567406176400, −7.54463706694837679097357205459, −6.35068227173636875972482030512, −4.67165229832493134016596280304, −4.26901520700210667846244787695, −0.945217057579748622313714488255, 2.32745663651917742497329581627, 4.44347302238199323016451924215, 5.71071836576077775830850192547, 6.75724365807925030726286782373, 7.888672098459935273545477497692, 9.163271679503416627936204475637, 10.68088568087414812493835505652, 11.36297092540353545696840418596, 12.02500556289803981418395006737, 13.05231547599016598040638329549

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