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Properties

Label 2-12e2-144.11-c1-0-15
Degree $2$
Conductor $144$
Sign $0.773 - 0.634i$
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  + (1.29 + 0.571i)2-s + (1.61 + 0.632i)3-s + (1.34 + 1.47i)4-s + (−3.81 − 1.02i)5-s + (1.72 + 1.73i)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 + (−2.65 + 0.710i)11-s + (1.23 + 3.23i)12-s + (−2.34 − 0.628i)13-s + (3.34 − 2.44i)14-s + (−5.50 − 4.05i)15-s + (−0.370 + 3.98i)16-s − 2.89i·17-s + ⋯
L(s)  = 1  + (0.914 + 0.404i)2-s + (0.931 + 0.364i)3-s + (0.673 + 0.739i)4-s + (−1.70 − 0.457i)5-s + (0.704 + 0.709i)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 + (−0.799 + 0.214i)11-s + (0.357 + 0.933i)12-s + (−0.651 − 0.174i)13-s + (0.895 − 0.654i)14-s + (−1.42 − 1.04i)15-s + (−0.0926 + 0.995i)16-s − 0.702i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.773 - 0.634i$
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: \((2,\ 144,\ (\ :1/2),\ 0.773 - 0.634i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77524 + 0.635156i\)
\(L(\frac12)\) \(\approx\) \(1.77524 + 0.635156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 - 0.571i)T \)
3 \( 1 + (-1.61 - 0.632i)T \)
good5 \( 1 + (3.81 + 1.02i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.46 + 2.54i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.65 - 0.710i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.34 + 0.628i)T + (11.2 + 6.5i)T^{2} \)
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 \( 1 + (-8.46 + 2.26i)T + (25.1 - 14.5i)T^{2} \)
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 \( 1 + (1.27 + 4.74i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.25 - 5.64i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.17 - 5.17i)T - 53iT^{2} \)
59 \( 1 + (-0.664 + 2.48i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.99 - 11.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.53 - 9.46i)T + (-58.0 - 33.5i)T^{2} \)
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 \( 1 + (3.37 + 12.5i)T + (-71.8 + 41.5i)T^{2} \)
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

−13.35544602894622044789372006403, −12.38921431530749556391083501039, −11.40478659877830966357023230878, −10.35769455047537554073299732531, −8.548601935709044981514318880098, −7.73017987938720358193325050357, −7.25245726897586127699843204300, −4.75235427763111666637388792024, −4.33382300673844370710336729015, −2.97727661325286391428365372281, 2.41033730974772165315687865063, 3.57342939362066295035939197062, 4.77668404615218155215088841665, 6.55725426144147564296018633789, 7.78191892564883643085714682129, 8.466373657827303956257798950755, 10.19549023746345024070051805137, 11.31401335283142057465920935182, 12.21451862341120209674797121093, 12.71578746998853462301854063135

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