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Properties

Label 2-12e2-144.13-c1-0-2
Degree $2$
Conductor $144$
Sign $-0.260 - 0.965i$
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  + (−1.31 + 0.521i)2-s + (0.241 + 1.71i)3-s + (1.45 − 1.37i)4-s + (0.531 + 1.98i)5-s + (−1.21 − 2.12i)6-s + (1.54 − 0.894i)7-s + (−1.19 + 2.56i)8-s + (−2.88 + 0.828i)9-s + (−1.73 − 2.33i)10-s + (2.58 + 0.693i)11-s + (2.70 + 2.16i)12-s + (−4.63 + 1.24i)13-s + (−1.57 + 1.98i)14-s + (−3.27 + 1.39i)15-s + (0.237 − 3.99i)16-s − 3.58·17-s + ⋯
L(s)  = 1  + (−0.929 + 0.368i)2-s + (0.139 + 0.990i)3-s + (0.727 − 0.685i)4-s + (0.237 + 0.887i)5-s + (−0.494 − 0.868i)6-s + (0.585 − 0.338i)7-s + (−0.423 + 0.905i)8-s + (−0.961 + 0.276i)9-s + (−0.548 − 0.737i)10-s + (0.779 + 0.208i)11-s + (0.780 + 0.625i)12-s + (−1.28 + 0.344i)13-s + (−0.419 + 0.530i)14-s + (−0.845 + 0.359i)15-s + (0.0594 − 0.998i)16-s − 0.870·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.260 - 0.965i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.260 - 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.489624 + 0.639483i\)
\(L(\frac12)\) \(\approx\) \(0.489624 + 0.639483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.31 - 0.521i)T \)
3 \( 1 + (-0.241 - 1.71i)T \)
good5 \( 1 + (-0.531 - 1.98i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.54 + 0.894i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.58 - 0.693i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.63 - 1.24i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + 3.58T + 17T^{2} \)
19 \( 1 + (-4.85 - 4.85i)T + 19iT^{2} \)
23 \( 1 + (0.446 + 0.257i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.72 + 6.44i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-4.05 + 7.01i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.25 + 1.25i)T - 37iT^{2} \)
41 \( 1 + (-4.07 - 2.35i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.57 - 1.76i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (3.48 + 6.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.26 + 5.26i)T - 53iT^{2} \)
59 \( 1 + (1.81 + 6.76i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.55 - 5.78i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.69 + 0.453i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.58iT - 71T^{2} \)
73 \( 1 - 12.5iT - 73T^{2} \)
79 \( 1 + (4.01 + 6.95i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.14 - 7.99i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 16.5iT - 89T^{2} \)
97 \( 1 + (4.15 + 7.20i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.07070926899200189312044927264, −11.79317022358397261836833265379, −11.14679485079460606330495316950, −9.974038919924770862048375882275, −9.615869312206760178719582641782, −8.219276264498720064211100140562, −7.16010336732257350331797553196, −5.92547210171219936016294379847, −4.40726399438576219086724944244, −2.47274917382598604253172003446, 1.19783216764023036779871761904, 2.71553236619217590691597736437, 5.05810358880221023135796736429, 6.69403613422980801898611199258, 7.66093607018003144606955707273, 8.816429728436979272417790809068, 9.270074493435492719033066969728, 10.91239432300900194528723985027, 11.96285555988256077540638237127, 12.44960465787859809159926793580

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