Properties

Label 2-12e2-144.131-c1-0-16
Degree $2$
Conductor $144$
Sign $0.652 + 0.757i$
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  + (1.25 − 0.659i)2-s + (−1.71 + 0.256i)3-s + (1.12 − 1.65i)4-s + (2.03 − 0.546i)5-s + (−1.97 + 1.45i)6-s + (−0.0638 − 0.110i)7-s + (0.323 − 2.80i)8-s + (2.86 − 0.878i)9-s + (2.18 − 2.02i)10-s + (−0.678 − 0.181i)11-s + (−1.51 + 3.11i)12-s + (−1.84 + 0.493i)13-s + (−0.152 − 0.0961i)14-s + (−3.35 + 1.45i)15-s + (−1.44 − 3.72i)16-s + 4.32i·17-s + ⋯
L(s)  = 1  + (0.884 − 0.466i)2-s + (−0.988 + 0.148i)3-s + (0.564 − 0.825i)4-s + (0.911 − 0.244i)5-s + (−0.805 + 0.592i)6-s + (−0.0241 − 0.0417i)7-s + (0.114 − 0.993i)8-s + (0.956 − 0.292i)9-s + (0.692 − 0.641i)10-s + (−0.204 − 0.0548i)11-s + (−0.436 + 0.899i)12-s + (−0.511 + 0.136i)13-s + (−0.0408 − 0.0256i)14-s + (−0.865 + 0.376i)15-s + (−0.362 − 0.932i)16-s + 1.04i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.652 + 0.757i$
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: \((2,\ 144,\ (\ :1/2),\ 0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35132 - 0.619170i\)
\(L(\frac12)\) \(\approx\) \(1.35132 - 0.619170i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.25 + 0.659i)T \)
3 \( 1 + (1.71 - 0.256i)T \)
good5 \( 1 + (-2.03 + 0.546i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.0638 + 0.110i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.678 + 0.181i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.84 - 0.493i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 4.32iT - 17T^{2} \)
19 \( 1 + (3.97 - 3.97i)T - 19iT^{2} \)
23 \( 1 + (-6.81 - 3.93i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.926 + 0.248i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.91 + 2.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.64 - 6.64i)T - 37iT^{2} \)
41 \( 1 + (-4.61 + 7.99i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.91 + 10.8i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-5.92 - 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.00259 - 0.00259i)T + 53iT^{2} \)
59 \( 1 + (1.09 + 4.09i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.19 - 4.44i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.538 + 2.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.80iT - 71T^{2} \)
73 \( 1 - 1.87iT - 73T^{2} \)
79 \( 1 + (-3.00 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.394 - 1.47i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (-3.31 - 5.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.82054147208121138906551012843, −12.17075097755400716864567818385, −10.91523363354215620897978913363, −10.30904646046915123920305674118, −9.256112577686912823605339763487, −7.15295025154467425278796702391, −5.94945609072208076286276511093, −5.29971583301881404192048212289, −3.93348134359786990578469187062, −1.79581985841554414074372007152, 2.52192766952375161552322216932, 4.63185691763076654078199909128, 5.51066813743960714096333144906, 6.59367927561188380312221809185, 7.36448630909583545160203965584, 9.132547071991120560033906640019, 10.54352269384399252102862732104, 11.31606329845326134256184652505, 12.53886190109080119213955979775, 13.10283431877529274132511493344

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