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Properties

Label 2-12e2-16.3-c2-0-14
Degree $2$
Conductor $144$
Sign $0.655 + 0.754i$
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  + (1.78 − 0.911i)2-s + (2.33 − 3.24i)4-s + (−1.00 + 1.00i)5-s + 10.0·7-s + (1.20 − 7.90i)8-s + (−0.875 + 2.71i)10-s + (−2.26 − 2.26i)11-s + (−6.88 − 6.88i)13-s + (17.8 − 9.13i)14-s + (−5.07 − 15.1i)16-s + 22.3·17-s + (−16.8 + 16.8i)19-s + (0.915 + 5.62i)20-s + (−6.09 − 1.96i)22-s − 33.2·23-s + ⋯
L(s)  = 1  + (0.890 − 0.455i)2-s + (0.584 − 0.811i)4-s + (−0.201 + 0.201i)5-s + 1.43·7-s + (0.150 − 0.988i)8-s + (−0.0875 + 0.271i)10-s + (−0.205 − 0.205i)11-s + (−0.529 − 0.529i)13-s + (1.27 − 0.652i)14-s + (−0.316 − 0.948i)16-s + 1.31·17-s + (−0.889 + 0.889i)19-s + (0.0457 + 0.281i)20-s + (−0.277 − 0.0894i)22-s − 1.44·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.655 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.655 + 0.754i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ 0.655 + 0.754i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.24429 - 1.02334i\)
\(L(\frac12)\) \(\approx\) \(2.24429 - 1.02334i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78 + 0.911i)T \)
3 \( 1 \)
good5 \( 1 + (1.00 - 1.00i)T - 25iT^{2} \)
7 \( 1 - 10.0T + 49T^{2} \)
11 \( 1 + (2.26 + 2.26i)T + 121iT^{2} \)
13 \( 1 + (6.88 + 6.88i)T + 169iT^{2} \)
17 \( 1 - 22.3T + 289T^{2} \)
19 \( 1 + (16.8 - 16.8i)T - 361iT^{2} \)
23 \( 1 + 33.2T + 529T^{2} \)
29 \( 1 + (-24.6 - 24.6i)T + 841iT^{2} \)
31 \( 1 - 41.3iT - 961T^{2} \)
37 \( 1 + (6.60 - 6.60i)T - 1.36e3iT^{2} \)
41 \( 1 + 47.1iT - 1.68e3T^{2} \)
43 \( 1 + (48.8 + 48.8i)T + 1.84e3iT^{2} \)
47 \( 1 - 45.6iT - 2.20e3T^{2} \)
53 \( 1 + (25.1 - 25.1i)T - 2.80e3iT^{2} \)
59 \( 1 + (6.23 + 6.23i)T + 3.48e3iT^{2} \)
61 \( 1 + (-35.9 - 35.9i)T + 3.72e3iT^{2} \)
67 \( 1 + (-10.2 + 10.2i)T - 4.48e3iT^{2} \)
71 \( 1 + 11.9T + 5.04e3T^{2} \)
73 \( 1 + 111. iT - 5.32e3T^{2} \)
79 \( 1 + 4.46iT - 6.24e3T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 21.9iT - 7.92e3T^{2} \)
97 \( 1 - 107.T + 9.40e3T^{2} \)
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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

−12.45368503921009534818124452911, −11.95737655115400107858833640132, −10.76856387561409349147339841093, −10.16625253058848583348422339637, −8.359781164880443851421843009511, −7.34871972003398701485825518481, −5.75582316581898929064531559277, −4.82558696578187842902816131349, −3.41864965395403729910714755149, −1.69491017288764090557004220601, 2.23407725220550549929032229678, 4.21623491613533889645173339489, 5.01057083568692735415638567865, 6.36156764574078866352997470779, 7.79379367935626944955094123774, 8.280581862329791243089597909348, 10.05575848726249632086470890771, 11.48226106428778770762916164191, 11.93083077749211224253090143741, 13.08538917883582524853210954768

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