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Properties

Label 2-12e2-16.13-c3-0-1
Degree $2$
Conductor $144$
Sign $-0.723 + 0.690i$
Analytic cond. $8.49627$
Root an. cond. $2.91483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.836 + 2.70i)2-s + (−6.59 − 4.52i)4-s + (0.596 − 0.596i)5-s + 29.0i·7-s + (17.7 − 14.0i)8-s + (1.11 + 2.11i)10-s + (−12.1 + 12.1i)11-s + (−48.5 − 48.5i)13-s + (−78.5 − 24.3i)14-s + (23.0 + 59.6i)16-s − 86.7·17-s + (−54.8 − 54.8i)19-s + (−6.63 + 1.23i)20-s + (−22.6 − 42.9i)22-s − 70.2i·23-s + ⋯
L(s)  = 1  + (−0.295 + 0.955i)2-s + (−0.824 − 0.565i)4-s + (0.0533 − 0.0533i)5-s + 1.57i·7-s + (0.784 − 0.620i)8-s + (0.0351 + 0.0667i)10-s + (−0.332 + 0.332i)11-s + (−1.03 − 1.03i)13-s + (−1.50 − 0.464i)14-s + (0.360 + 0.932i)16-s − 1.23·17-s + (−0.662 − 0.662i)19-s + (−0.0742 + 0.0138i)20-s + (−0.219 − 0.415i)22-s − 0.636i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.723 + 0.690i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ -0.723 + 0.690i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.122888 - 0.306804i\)
\(L(\frac12)\) \(\approx\) \(0.122888 - 0.306804i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.836 - 2.70i)T \)
3 \( 1 \)
good5 \( 1 + (-0.596 + 0.596i)T - 125iT^{2} \)
7 \( 1 - 29.0iT - 343T^{2} \)
11 \( 1 + (12.1 - 12.1i)T - 1.33e3iT^{2} \)
13 \( 1 + (48.5 + 48.5i)T + 2.19e3iT^{2} \)
17 \( 1 + 86.7T + 4.91e3T^{2} \)
19 \( 1 + (54.8 + 54.8i)T + 6.85e3iT^{2} \)
23 \( 1 + 70.2iT - 1.21e4T^{2} \)
29 \( 1 + (63.4 + 63.4i)T + 2.43e4iT^{2} \)
31 \( 1 + 8.86T + 2.97e4T^{2} \)
37 \( 1 + (21.7 - 21.7i)T - 5.06e4iT^{2} \)
41 \( 1 - 153. iT - 6.89e4T^{2} \)
43 \( 1 + (120. - 120. i)T - 7.95e4iT^{2} \)
47 \( 1 - 99.9T + 1.03e5T^{2} \)
53 \( 1 + (389. - 389. i)T - 1.48e5iT^{2} \)
59 \( 1 + (-324. + 324. i)T - 2.05e5iT^{2} \)
61 \( 1 + (0.339 + 0.339i)T + 2.26e5iT^{2} \)
67 \( 1 + (-565. - 565. i)T + 3.00e5iT^{2} \)
71 \( 1 - 419. iT - 3.57e5T^{2} \)
73 \( 1 - 374. iT - 3.89e5T^{2} \)
79 \( 1 + 705.T + 4.93e5T^{2} \)
83 \( 1 + (-947. - 947. i)T + 5.71e5iT^{2} \)
89 \( 1 - 4.72iT - 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−13.15264817584316250519641366173, −12.57849400076279769128458609762, −11.13447327317537142946203084439, −9.833123799666488150322761629909, −8.961443982864953745698945989092, −8.079985760130382447214046952452, −6.81738863993896901276198357864, −5.63750768836479377886624060105, −4.75112526599875212704105093873, −2.44156214852344736025109901273, 0.17013556599997181592509963257, 2.00863028108324565211242142770, 3.78111130701822142216656749183, 4.72226006421044931651469951109, 6.80752457948087710180298268082, 7.85380828801972223532556954367, 9.107523778661444126486448888949, 10.18541994627822770436552347203, 10.84237391707900060187326067121, 11.85121660786121657711881449117

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