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Properties

Label 2-12e2-144.11-c1-0-21
Degree $2$
Conductor $144$
Sign $-0.660 + 0.751i$
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  + (0.232 − 1.39i)2-s + (1.23 − 1.21i)3-s + (−1.89 − 0.649i)4-s + (−2.70 − 0.726i)5-s + (−1.40 − 2.00i)6-s + (−0.00424 + 0.00735i)7-s + (−1.34 + 2.48i)8-s + (0.0469 − 2.99i)9-s + (−1.64 + 3.61i)10-s + (3.00 − 0.804i)11-s + (−3.12 + 1.49i)12-s + (6.42 + 1.72i)13-s + (0.00927 + 0.00763i)14-s + (−4.22 + 2.39i)15-s + (3.15 + 2.45i)16-s − 3.37i·17-s + ⋯
L(s)  = 1  + (0.164 − 0.986i)2-s + (0.712 − 0.701i)3-s + (−0.945 − 0.324i)4-s + (−1.21 − 0.324i)5-s + (−0.574 − 0.818i)6-s + (−0.00160 + 0.00277i)7-s + (−0.476 + 0.879i)8-s + (0.0156 − 0.999i)9-s + (−0.519 + 1.14i)10-s + (0.904 − 0.242i)11-s + (−0.901 + 0.432i)12-s + (1.78 + 0.477i)13-s + (0.00247 + 0.00204i)14-s + (−1.09 + 0.618i)15-s + (0.789 + 0.614i)16-s − 0.819i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.660 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.660 + 0.751i$
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: \((2,\ 144,\ (\ :1/2),\ -0.660 + 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.484824 - 1.07134i\)
\(L(\frac12)\) \(\approx\) \(0.484824 - 1.07134i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.232 + 1.39i)T \)
3 \( 1 + (-1.23 + 1.21i)T \)
good5 \( 1 + (2.70 + 0.726i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.00424 - 0.00735i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.00 + 0.804i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-6.42 - 1.72i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.37iT - 17T^{2} \)
19 \( 1 + (-1.35 - 1.35i)T + 19iT^{2} \)
23 \( 1 + (5.38 - 3.11i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.27 + 0.878i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (6.70 - 3.86i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.769 - 0.769i)T + 37iT^{2} \)
41 \( 1 + (-2.64 - 4.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.60 - 5.99i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-0.0955 + 0.165i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.750 + 0.750i)T - 53iT^{2} \)
59 \( 1 + (2.43 - 9.08i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.64 + 9.85i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-2.08 + 7.77i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.02iT - 71T^{2} \)
73 \( 1 + 7.30iT - 73T^{2} \)
79 \( 1 + (4.36 + 2.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.35 - 12.5i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 18.1T + 89T^{2} \)
97 \( 1 + (-1.64 + 2.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.55420933991507279928486754516, −11.78517760365558949236644316775, −11.15801762513100985428661169374, −9.436952062804495532097869585650, −8.654782651227085421888880053160, −7.76238711618415633774658623470, −6.17486727913130174204057737429, −4.15105573366190069444807834958, −3.37639951278136888040660539487, −1.30482281412924884115292122135, 3.66484881578482014220812167983, 4.10400472288736726000441426720, 5.87885376100054461988609189219, 7.25755073640632011129866002118, 8.290351669272176770366372038433, 8.874843645119711761782077370478, 10.28023847503511727264601928342, 11.40437440566352826508116160136, 12.71693357336886268230623457160, 13.83962261484595276100038399483

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