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Properties

Label 2-12e2-144.11-c1-0-3
Degree $2$
Conductor $144$
Sign $0.949 - 0.312i$
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  + (−1.17 − 0.781i)2-s + (−1.69 − 0.377i)3-s + (0.779 + 1.84i)4-s + (1.05 + 0.282i)5-s + (1.69 + 1.76i)6-s + (−1.93 + 3.35i)7-s + (0.519 − 2.78i)8-s + (2.71 + 1.27i)9-s + (−1.02 − 1.15i)10-s + (3.53 − 0.946i)11-s + (−0.623 − 3.40i)12-s + (3.87 + 1.03i)13-s + (4.90 − 2.44i)14-s + (−1.67 − 0.874i)15-s + (−2.78 + 2.87i)16-s + 1.55i·17-s + ⋯
L(s)  = 1  + (−0.833 − 0.552i)2-s + (−0.975 − 0.217i)3-s + (0.389 + 0.920i)4-s + (0.471 + 0.126i)5-s + (0.693 + 0.720i)6-s + (−0.731 + 1.26i)7-s + (0.183 − 0.982i)8-s + (0.905 + 0.425i)9-s + (−0.323 − 0.365i)10-s + (1.06 − 0.285i)11-s + (−0.179 − 0.983i)12-s + (1.07 + 0.288i)13-s + (1.30 − 0.652i)14-s + (−0.432 − 0.225i)15-s + (−0.696 + 0.717i)16-s + 0.375i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.949 - 0.312i$
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: \((2,\ 144,\ (\ :1/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.593104 + 0.0951507i\)
\(L(\frac12)\) \(\approx\) \(0.593104 + 0.0951507i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.781i)T \)
3 \( 1 + (1.69 + 0.377i)T \)
good5 \( 1 + (-1.05 - 0.282i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.93 - 3.35i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.53 + 0.946i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.87 - 1.03i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 1.55iT - 17T^{2} \)
19 \( 1 + (-4.06 - 4.06i)T + 19iT^{2} \)
23 \( 1 + (3.86 - 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.28 - 1.14i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.85 + 1.07i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.04 - 6.04i)T + 37iT^{2} \)
41 \( 1 + (-1.59 - 2.76i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.47 + 5.51i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.0494 + 0.0856i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.72 - 1.72i)T - 53iT^{2} \)
59 \( 1 + (-3.58 + 13.3i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.33 + 8.69i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.251 + 0.939i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.11iT - 71T^{2} \)
73 \( 1 + 10.4iT - 73T^{2} \)
79 \( 1 + (-14.2 - 8.22i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.02 + 15.0i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-0.0532 + 0.0922i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.73442915287706135656199631897, −11.88382244326013001122945147324, −11.32244128272959854095083717386, −9.969602298812102232819398127837, −9.330664350443292603185051199189, −8.068455509000181552567592595066, −6.44644129476791035175554554144, −5.91026477085864467098922840123, −3.66428209660823062158283591773, −1.71653242876734001482237247533, 1.00039119801692588905605276178, 4.11011804842792654002709711273, 5.72246307873400240408024643736, 6.60087584035467889420130979538, 7.46796062164613458684710606277, 9.244564386924071857436407533324, 9.865050360365248568233729925157, 10.82572680213732750517145852040, 11.67199709492084424912681478746, 13.19429610246927061736694938622

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