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Properties

Label 2-12e2-144.83-c1-0-11
Degree $2$
Conductor $144$
Sign $0.819 - 0.573i$
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  + (1.14 + 0.834i)2-s + (0.632 − 1.61i)3-s + (0.606 + 1.90i)4-s + (−1.02 + 3.81i)5-s + (2.06 − 1.31i)6-s + (1.46 − 2.54i)7-s + (−0.897 + 2.68i)8-s + (−2.20 − 2.03i)9-s + (−4.35 + 3.50i)10-s + (−0.710 − 2.65i)11-s + (3.45 + 0.226i)12-s + (0.628 − 2.34i)13-s + (3.79 − 1.67i)14-s + (5.50 + 4.05i)15-s + (−3.26 + 2.31i)16-s − 2.89i·17-s + ⋯
L(s)  = 1  + (0.807 + 0.590i)2-s + (0.364 − 0.931i)3-s + (0.303 + 0.952i)4-s + (−0.457 + 1.70i)5-s + (0.844 − 0.536i)6-s + (0.554 − 0.960i)7-s + (−0.317 + 0.948i)8-s + (−0.733 − 0.679i)9-s + (−1.37 + 1.10i)10-s + (−0.214 − 0.799i)11-s + (0.997 + 0.0653i)12-s + (0.174 − 0.651i)13-s + (1.01 − 0.448i)14-s + (1.42 + 1.04i)15-s + (−0.815 + 0.578i)16-s − 0.702i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.819 - 0.573i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63222 + 0.514809i\)
\(L(\frac12)\) \(\approx\) \(1.63222 + 0.514809i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.14 - 0.834i)T \)
3 \( 1 + (-0.632 + 1.61i)T \)
good5 \( 1 + (1.02 - 3.81i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.46 + 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.710 + 2.65i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.628 + 2.34i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + 2.89iT - 17T^{2} \)
19 \( 1 + (1.99 - 1.99i)T - 19iT^{2} \)
23 \( 1 + (2.07 - 1.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.26 - 8.46i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.439 + 0.253i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.36 + 1.36i)T - 37iT^{2} \)
41 \( 1 + (0.745 + 1.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.74 + 1.27i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-3.25 + 5.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.17 - 5.17i)T + 53iT^{2} \)
59 \( 1 + (-2.48 - 0.664i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (11.1 - 2.99i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-9.46 - 2.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.65iT - 71T^{2} \)
73 \( 1 - 4.91iT - 73T^{2} \)
79 \( 1 + (-3.61 - 2.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (12.5 - 3.37i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 7.33T + 89T^{2} \)
97 \( 1 + (-2.50 + 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.65178506995582719342813152106, −12.35045135375655695939308095578, −11.28007551528066453744601072127, −10.61889584887508890493095431983, −8.400570275424517605841494490201, −7.50208672362651900444025722156, −6.95471513746648002374808758969, −5.81715011057060988404818156301, −3.77462491511002858624186580239, −2.79182795520103556129576205784, 2.13214046323816111084363269738, 4.20380276423380083377550748337, 4.72438507338510932288700577371, 5.81484043281258221716083337577, 8.148763260299675989311229400310, 9.006370950721565448735609217979, 9.876490201201609506744689479335, 11.27025360219602790454857694677, 12.06541502856544188525758412299, 12.83731304288650671397569692595

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