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Properties

Label 2-12e2-144.59-c1-0-17
Degree $2$
Conductor $144$
Sign $0.986 + 0.163i$
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  + (1.24 + 0.664i)2-s + (0.517 − 1.65i)3-s + (1.11 + 1.65i)4-s + (−0.521 − 1.94i)5-s + (1.74 − 1.71i)6-s + (−0.322 − 0.558i)7-s + (0.290 + 2.81i)8-s + (−2.46 − 1.71i)9-s + (0.642 − 2.77i)10-s + (−1.49 + 5.59i)11-s + (3.32 − 0.986i)12-s + (0.530 + 1.97i)13-s + (−0.0311 − 0.910i)14-s + (−3.48 − 0.144i)15-s + (−1.50 + 3.70i)16-s − 3.05i·17-s + ⋯
L(s)  = 1  + (0.882 + 0.470i)2-s + (0.298 − 0.954i)3-s + (0.558 + 0.829i)4-s + (−0.233 − 0.869i)5-s + (0.712 − 0.701i)6-s + (−0.121 − 0.210i)7-s + (0.102 + 0.994i)8-s + (−0.821 − 0.570i)9-s + (0.203 − 0.877i)10-s + (−0.451 + 1.68i)11-s + (0.958 − 0.284i)12-s + (0.147 + 0.548i)13-s + (−0.00833 − 0.243i)14-s + (−0.899 − 0.0373i)15-s + (−0.377 + 0.926i)16-s − 0.741i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.986 + 0.163i$
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: \((2,\ 144,\ (\ :1/2),\ 0.986 + 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75348 - 0.144361i\)
\(L(\frac12)\) \(\approx\) \(1.75348 - 0.144361i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.664i)T \)
3 \( 1 + (-0.517 + 1.65i)T \)
good5 \( 1 + (0.521 + 1.94i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.322 + 0.558i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.49 - 5.59i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.530 - 1.97i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + 3.05iT - 17T^{2} \)
19 \( 1 + (-4.11 - 4.11i)T + 19iT^{2} \)
23 \( 1 + (7.05 + 4.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.20 + 4.49i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (0.147 + 0.0854i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.65 + 2.65i)T + 37iT^{2} \)
41 \( 1 + (0.983 - 1.70i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.821 - 0.220i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.02 + 6.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.25 + 3.25i)T - 53iT^{2} \)
59 \( 1 + (-0.506 + 0.135i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.00 - 0.805i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.17 - 0.583i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 - 9.33iT - 73T^{2} \)
79 \( 1 + (-9.44 + 5.45i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.3 - 3.31i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 3.86T + 89T^{2} \)
97 \( 1 + (7.33 + 12.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.08023718879704430769617363133, −12.20637707162695909557612313858, −11.84973499357915787511171492463, −9.894627587011534139467563320226, −8.456566365193760719174438878905, −7.59064288612886943685827110605, −6.69955067484367312933139707781, −5.29393097860765272348012110655, −4.06276306051349242329185740427, −2.17385203792850702732016033906, 2.94124570498055163400936717707, 3.58604206715700182600149356328, 5.24914668080165457660353331595, 6.18597548042749552213525639282, 7.87425241669611631243059768271, 9.253279717074005223573062028704, 10.56044070911277543371345197280, 10.91739593114521456361662821250, 11.93535909874819844818001440983, 13.46465494868199436608834734604

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