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Properties

Label 2-12e2-144.131-c1-0-6
Degree $2$
Conductor $144$
Sign $0.991 + 0.132i$
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  + (−1.07 − 0.916i)2-s + (1.67 + 0.427i)3-s + (0.320 + 1.97i)4-s + (−0.170 + 0.0458i)5-s + (−1.41 − 1.99i)6-s + (1.17 + 2.03i)7-s + (1.46 − 2.42i)8-s + (2.63 + 1.43i)9-s + (0.226 + 0.107i)10-s + (−0.340 − 0.0913i)11-s + (−0.305 + 3.45i)12-s + (−1.49 + 0.399i)13-s + (0.598 − 3.26i)14-s + (−0.306 + 0.00382i)15-s + (−3.79 + 1.26i)16-s − 3.58i·17-s + ⋯
L(s)  = 1  + (−0.761 − 0.647i)2-s + (0.969 + 0.246i)3-s + (0.160 + 0.987i)4-s + (−0.0764 + 0.0204i)5-s + (−0.578 − 0.815i)6-s + (0.443 + 0.768i)7-s + (0.517 − 0.855i)8-s + (0.878 + 0.478i)9-s + (0.0715 + 0.0339i)10-s + (−0.102 − 0.0275i)11-s + (−0.0880 + 0.996i)12-s + (−0.413 + 0.110i)13-s + (0.160 − 0.873i)14-s + (−0.0791 + 0.000988i)15-s + (−0.948 + 0.316i)16-s − 0.868i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.991 + 0.132i$
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: \((2,\ 144,\ (\ :1/2),\ 0.991 + 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05864 - 0.0703719i\)
\(L(\frac12)\) \(\approx\) \(1.05864 - 0.0703719i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 + 0.916i)T \)
3 \( 1 + (-1.67 - 0.427i)T \)
good5 \( 1 + (0.170 - 0.0458i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.17 - 2.03i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.340 + 0.0913i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.49 - 0.399i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 3.58iT - 17T^{2} \)
19 \( 1 + (-5.36 + 5.36i)T - 19iT^{2} \)
23 \( 1 + (-0.165 - 0.0953i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (9.10 + 2.43i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (3.43 + 1.98i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.28 - 3.28i)T - 37iT^{2} \)
41 \( 1 + (4.25 - 7.37i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.09 + 4.09i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-4.93 - 8.53i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.83 + 4.83i)T + 53iT^{2} \)
59 \( 1 + (0.720 + 2.68i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.13 + 7.97i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.06 + 11.4i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.13iT - 71T^{2} \)
73 \( 1 - 5.67iT - 73T^{2} \)
79 \( 1 + (-12.8 + 7.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.31 - 12.3i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 - 3.05T + 89T^{2} \)
97 \( 1 + (0.996 + 1.72i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.05850556389694420787965853921, −11.83294173994983227124013242282, −11.08318753361510630391886489279, −9.550159359149155262550260104818, −9.277740402699432728755930817001, −8.003498210119450136105467790288, −7.23364026890578226585867385603, −4.97746858154872983542807247827, −3.35550892050438307896237433282, −2.12382356680289453530598186440, 1.70112376159042157314701772551, 3.88255319808369016891371714270, 5.62350172824571111955628803289, 7.24889233642238225958545710451, 7.73044010056297975789014413082, 8.797861062741152124770885230439, 9.868997182241823847297747865005, 10.67742863554048508224456721445, 12.15631270372261387263129552336, 13.50287055351023921067931358826

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