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Properties

Label 2-12e2-144.11-c1-0-2
Degree $2$
Conductor $144$
Sign $-0.462 - 0.886i$
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  + (−0.665 − 1.24i)2-s + (0.117 + 1.72i)3-s + (−1.11 + 1.66i)4-s + (−3.58 − 0.961i)5-s + (2.07 − 1.29i)6-s + (−1.29 + 2.23i)7-s + (2.81 + 0.286i)8-s + (−2.97 + 0.406i)9-s + (1.18 + 5.11i)10-s + (−2.02 + 0.541i)11-s + (−3.00 − 1.73i)12-s + (−0.998 − 0.267i)13-s + (3.65 + 0.123i)14-s + (1.23 − 6.31i)15-s + (−1.51 − 3.70i)16-s + 4.13i·17-s + ⋯
L(s)  = 1  + (−0.470 − 0.882i)2-s + (0.0679 + 0.997i)3-s + (−0.557 + 0.830i)4-s + (−1.60 − 0.430i)5-s + (0.848 − 0.529i)6-s + (−0.488 + 0.845i)7-s + (0.994 + 0.101i)8-s + (−0.990 + 0.135i)9-s + (0.375 + 1.61i)10-s + (−0.609 + 0.163i)11-s + (−0.866 − 0.499i)12-s + (−0.276 − 0.0741i)13-s + (0.976 + 0.0330i)14-s + (0.319 − 1.63i)15-s + (−0.378 − 0.925i)16-s + 1.00i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 - 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.462 - 0.886i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.462 - 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.166995 + 0.275429i\)
\(L(\frac12)\) \(\approx\) \(0.166995 + 0.275429i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.665 + 1.24i)T \)
3 \( 1 + (-0.117 - 1.72i)T \)
good5 \( 1 + (3.58 + 0.961i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.29 - 2.23i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.02 - 0.541i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.998 + 0.267i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 4.13iT - 17T^{2} \)
19 \( 1 + (-2.49 - 2.49i)T + 19iT^{2} \)
23 \( 1 + (-7.01 + 4.05i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.02 - 1.88i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.03 - 1.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.75 + 4.75i)T + 37iT^{2} \)
41 \( 1 + (-0.636 - 1.10i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.389 - 1.45i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.60 - 6.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.546 + 0.546i)T - 53iT^{2} \)
59 \( 1 + (2.14 - 8.00i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.81 - 6.77i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.751 + 2.80i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 6.59iT - 71T^{2} \)
73 \( 1 + 8.78iT - 73T^{2} \)
79 \( 1 + (-5.64 - 3.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.06 - 7.71i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 14.1T + 89T^{2} \)
97 \( 1 + (-6.54 + 11.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.85438048123317652137714625885, −12.29565971467777535703003668918, −11.28116139217515144644138760815, −10.53112430580533662029667072921, −9.256263011934011843116053646328, −8.563991906211554934910838971057, −7.58182943371596355476738038433, −5.23209098403542634629574348624, −4.03804612281982287075399555614, −3.01516186031423621831627930050, 0.36757874302497931361126040671, 3.39913447235785467502062482041, 5.15706970630585195890090749917, 6.99851470314143108547607399013, 7.22804932192823878262990069813, 8.113554105717226965193220640677, 9.381057004667009108945393846856, 10.89134083998055405682203540935, 11.62712876087257476838268215543, 13.05126450013568371628755819747

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