Properties

Label 2-12e2-144.131-c1-0-9
Degree $2$
Conductor $144$
Sign $0.714 + 0.699i$
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  + (−1.40 + 0.165i)2-s + (1.03 − 1.38i)3-s + (1.94 − 0.463i)4-s + (1.15 − 0.310i)5-s + (−1.23 + 2.11i)6-s + (0.356 + 0.616i)7-s + (−2.65 + 0.972i)8-s + (−0.840 − 2.87i)9-s + (−1.57 + 0.627i)10-s + (−2.28 − 0.611i)11-s + (1.37 − 3.17i)12-s + (4.90 − 1.31i)13-s + (−0.601 − 0.807i)14-s + (0.773 − 1.92i)15-s + (3.57 − 1.80i)16-s + 0.863i·17-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)2-s + (0.599 − 0.800i)3-s + (0.972 − 0.231i)4-s + (0.517 − 0.138i)5-s + (−0.502 + 0.864i)6-s + (0.134 + 0.233i)7-s + (−0.939 + 0.343i)8-s + (−0.280 − 0.959i)9-s + (−0.498 + 0.198i)10-s + (−0.687 − 0.184i)11-s + (0.398 − 0.917i)12-s + (1.36 − 0.364i)13-s + (−0.160 − 0.215i)14-s + (0.199 − 0.497i)15-s + (0.892 − 0.450i)16-s + 0.209i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 + 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.714 + 0.699i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.714 + 0.699i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.872609 - 0.356306i\)
\(L(\frac12)\) \(\approx\) \(0.872609 - 0.356306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.165i)T \)
3 \( 1 + (-1.03 + 1.38i)T \)
good5 \( 1 + (-1.15 + 0.310i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.356 - 0.616i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.28 + 0.611i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-4.90 + 1.31i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 0.863iT - 17T^{2} \)
19 \( 1 + (0.539 - 0.539i)T - 19iT^{2} \)
23 \( 1 + (0.689 + 0.398i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.31 - 2.22i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.95 - 6.95i)T - 37iT^{2} \)
41 \( 1 + (3.17 - 5.49i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.22 - 12.0i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.87 - 8.87i)T + 53iT^{2} \)
59 \( 1 + (3.40 + 12.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.146 - 0.548i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.84 - 6.89i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.03iT - 71T^{2} \)
73 \( 1 + 11.6iT - 73T^{2} \)
79 \( 1 + (0.841 - 0.486i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.99 - 11.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 4.35T + 89T^{2} \)
97 \( 1 + (2.89 + 5.01i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.03619623954444828168945297673, −11.91403937982697725678221951613, −10.79346183286225871754862414503, −9.700385973552209800570774182401, −8.549600091385199942102382275448, −8.054090003228239500663549922049, −6.66338766067698220738286345052, −5.73622813436928917870020721821, −3.05457870911800453054223513901, −1.52596933061496290350472427046, 2.20041533735773751947474382540, 3.74819633168144509696347433223, 5.60815715154860426832392318133, 7.08550601283593515599870878043, 8.369055307480080378035049827050, 9.032058866290851376994759901116, 10.29188781281688409343030006209, 10.62107703872480506936550258939, 11.90615205976484695402405072436, 13.43225981029895579142025943052

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