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Properties

Label 2-12e2-144.59-c1-0-13
Degree $2$
Conductor $144$
Sign $0.931 + 0.363i$
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  + (−0.503 + 1.32i)2-s + (1.05 − 1.36i)3-s + (−1.49 − 1.32i)4-s + (−0.746 − 2.78i)5-s + (1.27 + 2.09i)6-s + (1.16 + 2.02i)7-s + (2.50 − 1.30i)8-s + (−0.753 − 2.90i)9-s + (4.05 + 0.415i)10-s + (1.48 − 5.53i)11-s + (−3.40 + 0.636i)12-s + (1.04 + 3.90i)13-s + (−3.25 + 0.525i)14-s + (−4.60 − 1.93i)15-s + (0.462 + 3.97i)16-s + 6.45i·17-s + ⋯
L(s)  = 1  + (−0.355 + 0.934i)2-s + (0.611 − 0.790i)3-s + (−0.746 − 0.664i)4-s + (−0.333 − 1.24i)5-s + (0.521 + 0.853i)6-s + (0.440 + 0.763i)7-s + (0.887 − 0.461i)8-s + (−0.251 − 0.967i)9-s + (1.28 + 0.131i)10-s + (0.447 − 1.67i)11-s + (−0.982 + 0.183i)12-s + (0.290 + 1.08i)13-s + (−0.870 + 0.140i)14-s + (−1.19 − 0.498i)15-s + (0.115 + 0.993i)16-s + 1.56i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01708 - 0.191446i\)
\(L(\frac12)\) \(\approx\) \(1.01708 - 0.191446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.503 - 1.32i)T \)
3 \( 1 + (-1.05 + 1.36i)T \)
good5 \( 1 + (0.746 + 2.78i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.16 - 2.02i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.48 + 5.53i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.04 - 3.90i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.45iT - 17T^{2} \)
19 \( 1 + (1.50 + 1.50i)T + 19iT^{2} \)
23 \( 1 + (0.0418 + 0.0241i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.36 - 5.08i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.65 - 0.952i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.489 - 0.489i)T + 37iT^{2} \)
41 \( 1 + (0.0155 - 0.0269i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.80 - 1.01i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.0913 + 0.158i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.62 - 6.62i)T - 53iT^{2} \)
59 \( 1 + (-4.15 + 1.11i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.39 - 1.71i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.808 - 0.216i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.04iT - 71T^{2} \)
73 \( 1 - 4.74iT - 73T^{2} \)
79 \( 1 + (7.29 - 4.21i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.9 - 3.20i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.85T + 89T^{2} \)
97 \( 1 + (-3.29 - 5.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.20554224656903833151721819552, −12.31104333672930485812862960150, −11.13413753142546665025744040139, −9.147212679382925365972370575606, −8.622587437690545843810201170153, −8.177121426540389189276170925683, −6.57769428093282757621511335334, −5.62338732523614934228929932905, −4.03970242784843373682575532507, −1.35775974278513667819909933116, 2.47559919869775809766360085439, 3.70857688778951841202241971661, 4.74937260458185085873902167437, 7.25258035223192959221234695421, 7.921862601811510825553659824667, 9.458502469642054968192989623564, 10.18390406074603510574111122252, 10.88856117091091545746023948340, 11.80923250326056028246671296206, 13.20101420107095763857285372202

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