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Properties

Label 2-12e2-144.13-c1-0-21
Degree $2$
Conductor $144$
Sign $-0.987 - 0.156i$
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  + (−0.709 − 1.22i)2-s + (−0.705 − 1.58i)3-s + (−0.993 + 1.73i)4-s + (−0.679 − 2.53i)5-s + (−1.43 + 1.98i)6-s + (−0.614 + 0.354i)7-s + (2.82 − 0.0162i)8-s + (−2.00 + 2.23i)9-s + (−2.62 + 2.63i)10-s + (−3.63 − 0.973i)11-s + (3.44 + 0.347i)12-s + (−0.519 + 0.139i)13-s + (0.869 + 0.499i)14-s + (−3.53 + 2.86i)15-s + (−2.02 − 3.44i)16-s + 6.08·17-s + ⋯
L(s)  = 1  + (−0.501 − 0.865i)2-s + (−0.407 − 0.913i)3-s + (−0.496 + 0.867i)4-s + (−0.303 − 1.13i)5-s + (−0.585 + 0.810i)6-s + (−0.232 + 0.134i)7-s + (0.999 − 0.00574i)8-s + (−0.668 + 0.743i)9-s + (−0.828 + 0.831i)10-s + (−1.09 − 0.293i)11-s + (0.994 + 0.100i)12-s + (−0.143 + 0.0385i)13-s + (0.232 + 0.133i)14-s + (−0.912 + 0.739i)15-s + (−0.506 − 0.862i)16-s + 1.47·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.987 - 0.156i$
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: \((2,\ 144,\ (\ :1/2),\ -0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0407347 + 0.516472i\)
\(L(\frac12)\) \(\approx\) \(0.0407347 + 0.516472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.709 + 1.22i)T \)
3 \( 1 + (0.705 + 1.58i)T \)
good5 \( 1 + (0.679 + 2.53i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.614 - 0.354i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.63 + 0.973i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.519 - 0.139i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 6.08T + 17T^{2} \)
19 \( 1 + (1.86 + 1.86i)T + 19iT^{2} \)
23 \( 1 + (4.94 + 2.85i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.59 + 9.68i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-2.14 + 3.70i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 + 3.75i)T - 37iT^{2} \)
41 \( 1 + (-1.57 - 0.906i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.62 - 2.31i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.95 + 5.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.56 - 8.56i)T - 53iT^{2} \)
59 \( 1 + (-1.39 - 5.19i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.175 - 0.655i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-7.33 + 1.96i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.51iT - 71T^{2} \)
73 \( 1 + 7.36iT - 73T^{2} \)
79 \( 1 + (-0.0143 - 0.0248i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.00 - 14.9i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.86iT - 89T^{2} \)
97 \( 1 + (-5.66 - 9.80i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.47099418463183214021694290270, −11.80530226848553585597476894786, −10.66309791226174967900330415636, −9.518150991496020617474915831227, −8.150988531132924497245678699434, −7.81735019712849689482936724195, −5.87959100538738330397407723164, −4.52892429384363134115635239354, −2.53401140623273835763606728147, −0.64337888643555313222078944267, 3.36631809577162202491171509547, 4.99294756456899978182485423404, 6.08732254798783012586716960108, 7.25424976701873630060360301027, 8.281344204662287949405186604307, 9.843975275905378012017580308589, 10.27700771534151135631592758977, 11.16070948687935963037535541859, 12.60166960642431952850757906490, 14.31593696317316060153180593078

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