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Properties

Label 2-12e2-48.5-c2-0-2
Degree $2$
Conductor $144$
Sign $-0.340 - 0.940i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.99 − 0.107i)2-s + (3.97 + 0.430i)4-s + (−1.92 + 1.92i)5-s + 2.43i·7-s + (−7.89 − 1.28i)8-s + (4.05 − 3.64i)10-s + (−2.79 + 2.79i)11-s + (−1.54 + 1.54i)13-s + (0.261 − 4.85i)14-s + (15.6 + 3.42i)16-s + 20.3i·17-s + (−25.0 + 25.0i)19-s + (−8.49 + 6.83i)20-s + (5.87 − 5.27i)22-s − 3.55·23-s + ⋯
L(s)  = 1  + (−0.998 − 0.0538i)2-s + (0.994 + 0.107i)4-s + (−0.385 + 0.385i)5-s + 0.347i·7-s + (−0.986 − 0.160i)8-s + (0.405 − 0.364i)10-s + (−0.253 + 0.253i)11-s + (−0.118 + 0.118i)13-s + (0.0187 − 0.346i)14-s + (0.976 + 0.213i)16-s + 1.19i·17-s + (−1.31 + 1.31i)19-s + (−0.424 + 0.341i)20-s + (0.267 − 0.239i)22-s − 0.154·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.340 - 0.940i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.340 - 0.940i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.339394 + 0.483702i\)
\(L(\frac12)\) \(\approx\) \(0.339394 + 0.483702i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 + 0.107i)T \)
3 \( 1 \)
good5 \( 1 + (1.92 - 1.92i)T - 25iT^{2} \)
7 \( 1 - 2.43iT - 49T^{2} \)
11 \( 1 + (2.79 - 2.79i)T - 121iT^{2} \)
13 \( 1 + (1.54 - 1.54i)T - 169iT^{2} \)
17 \( 1 - 20.3iT - 289T^{2} \)
19 \( 1 + (25.0 - 25.0i)T - 361iT^{2} \)
23 \( 1 + 3.55T + 529T^{2} \)
29 \( 1 + (-23.4 - 23.4i)T + 841iT^{2} \)
31 \( 1 + 13.3T + 961T^{2} \)
37 \( 1 + (25.4 + 25.4i)T + 1.36e3iT^{2} \)
41 \( 1 - 64.0T + 1.68e3T^{2} \)
43 \( 1 + (24.6 + 24.6i)T + 1.84e3iT^{2} \)
47 \( 1 + 79.5iT - 2.20e3T^{2} \)
53 \( 1 + (39.8 - 39.8i)T - 2.80e3iT^{2} \)
59 \( 1 + (-19.2 + 19.2i)T - 3.48e3iT^{2} \)
61 \( 1 + (-63.5 + 63.5i)T - 3.72e3iT^{2} \)
67 \( 1 + (65.1 - 65.1i)T - 4.48e3iT^{2} \)
71 \( 1 + 84.6T + 5.04e3T^{2} \)
73 \( 1 - 39.7iT - 5.32e3T^{2} \)
79 \( 1 - 109.T + 6.24e3T^{2} \)
83 \( 1 + (-14.7 - 14.7i)T + 6.88e3iT^{2} \)
89 \( 1 - 32.3T + 7.92e3T^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−12.77016448863944942000893999418, −12.07153726942941446055854847729, −10.83010817876128453966052501087, −10.25691234411509477062126916529, −8.907981094146104227232250104736, −8.066033990144926730568536872458, −6.96075554233969876558685403951, −5.80544584072742587418576706984, −3.70794799595472507798512285031, −2.00232820419904843358087011240, 0.51117120098800982783037786519, 2.66520759574843113838527661044, 4.62705870220748228415155771676, 6.30104393767953145218528444052, 7.42071360814966414923083787165, 8.406324961629745141314936074270, 9.339425516226441907719848585849, 10.47395637667540597910739136389, 11.34874374095688860793627732403, 12.31102960844604653318868237809

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