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Properties

Label 2-12e2-48.5-c2-0-6
Degree $2$
Conductor $144$
Sign $0.264 - 0.964i$
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  + (1.95 + 0.412i)2-s + (3.66 + 1.61i)4-s + (−5.52 + 5.52i)5-s + 7.79i·7-s + (6.49 + 4.66i)8-s + (−13.0 + 8.53i)10-s + (2.48 − 2.48i)11-s + (13.3 − 13.3i)13-s + (−3.21 + 15.2i)14-s + (10.7 + 11.8i)16-s − 5.86i·17-s + (18.5 − 18.5i)19-s + (−29.1 + 11.3i)20-s + (5.88 − 3.83i)22-s − 34.3·23-s + ⋯
L(s)  = 1  + (0.978 + 0.206i)2-s + (0.915 + 0.403i)4-s + (−1.10 + 1.10i)5-s + 1.11i·7-s + (0.812 + 0.583i)8-s + (−1.30 + 0.853i)10-s + (0.225 − 0.225i)11-s + (1.02 − 1.02i)13-s + (−0.229 + 1.08i)14-s + (0.674 + 0.738i)16-s − 0.344i·17-s + (0.977 − 0.977i)19-s + (−1.45 + 0.565i)20-s + (0.267 − 0.174i)22-s − 1.49·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.264 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.264 - 0.964i$
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: \((2,\ 144,\ (\ :1),\ 0.264 - 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.76875 + 1.34861i\)
\(L(\frac12)\) \(\approx\) \(1.76875 + 1.34861i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 - 0.412i)T \)
3 \( 1 \)
good5 \( 1 + (5.52 - 5.52i)T - 25iT^{2} \)
7 \( 1 - 7.79iT - 49T^{2} \)
11 \( 1 + (-2.48 + 2.48i)T - 121iT^{2} \)
13 \( 1 + (-13.3 + 13.3i)T - 169iT^{2} \)
17 \( 1 + 5.86iT - 289T^{2} \)
19 \( 1 + (-18.5 + 18.5i)T - 361iT^{2} \)
23 \( 1 + 34.3T + 529T^{2} \)
29 \( 1 + (-21.4 - 21.4i)T + 841iT^{2} \)
31 \( 1 + 30.6T + 961T^{2} \)
37 \( 1 + (-30.3 - 30.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 3.12T + 1.68e3T^{2} \)
43 \( 1 + (9.94 + 9.94i)T + 1.84e3iT^{2} \)
47 \( 1 + 38.4iT - 2.20e3T^{2} \)
53 \( 1 + (-61.1 + 61.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (-2.98 + 2.98i)T - 3.48e3iT^{2} \)
61 \( 1 + (-3.88 + 3.88i)T - 3.72e3iT^{2} \)
67 \( 1 + (47.0 - 47.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 97.5T + 5.04e3T^{2} \)
73 \( 1 + 106. iT - 5.32e3T^{2} \)
79 \( 1 + 96.8T + 6.24e3T^{2} \)
83 \( 1 + (88.8 + 88.8i)T + 6.88e3iT^{2} \)
89 \( 1 + 54.8T + 7.92e3T^{2} \)
97 \( 1 + 5.00T + 9.40e3T^{2} \)
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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

−13.10713324879367216272597275040, −11.86817692487165250550345317788, −11.49117384086570763525762975184, −10.42150568585131407780035992916, −8.558437529756956589320482667046, −7.57204852114356492432965824699, −6.46892975925908230610863400029, −5.37383689574553242473379280015, −3.70253168366346034956611704042, −2.79508496998583247688213947980, 1.27562827434977983264651602548, 3.96988704261786854450511180016, 4.18396664746638421684335790251, 5.85834992855798284170093200743, 7.26570965756013539478297166219, 8.199682424315219902200306113533, 9.736009456045815536250463742561, 11.01069475860837723927048295387, 11.84934031790342331662322654075, 12.57038987610863178204311219868

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