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Properties

Label 2-12e2-16.11-c2-0-5
Degree $2$
Conductor $144$
Sign $-0.368 - 0.929i$
Analytic cond. $3.92371$
Root an. cond. $1.98083$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 1.66i)2-s + (−1.55 + 3.68i)4-s + (4.23 + 4.23i)5-s − 0.262·7-s + (−7.86 + 1.46i)8-s + (−2.38 + 11.7i)10-s + (8.60 − 8.60i)11-s + (−15.9 + 15.9i)13-s + (−0.289 − 0.437i)14-s + (−11.1 − 11.4i)16-s + 3.51·17-s + (10.7 + 10.7i)19-s + (−22.2 + 9.00i)20-s + (23.8 + 4.84i)22-s + 16.4·23-s + ⋯
L(s)  = 1  + (0.552 + 0.833i)2-s + (−0.389 + 0.920i)4-s + (0.847 + 0.847i)5-s − 0.0374·7-s + (−0.983 + 0.183i)8-s + (−0.238 + 1.17i)10-s + (0.782 − 0.782i)11-s + (−1.23 + 1.23i)13-s + (−0.0206 − 0.0312i)14-s + (−0.695 − 0.718i)16-s + 0.206·17-s + (0.566 + 0.566i)19-s + (−1.11 + 0.450i)20-s + (1.08 + 0.220i)22-s + 0.717·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.368 - 0.929i$
Analytic conductor: \(3.92371\)
Root analytic conductor: \(1.98083\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1),\ -0.368 - 0.929i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.10162 + 1.62094i\)
\(L(\frac12)\) \(\approx\) \(1.10162 + 1.62094i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 1.66i)T \)
3 \( 1 \)
good5 \( 1 + (-4.23 - 4.23i)T + 25iT^{2} \)
7 \( 1 + 0.262T + 49T^{2} \)
11 \( 1 + (-8.60 + 8.60i)T - 121iT^{2} \)
13 \( 1 + (15.9 - 15.9i)T - 169iT^{2} \)
17 \( 1 - 3.51T + 289T^{2} \)
19 \( 1 + (-10.7 - 10.7i)T + 361iT^{2} \)
23 \( 1 - 16.4T + 529T^{2} \)
29 \( 1 + (-25.9 + 25.9i)T - 841iT^{2} \)
31 \( 1 + 46.2iT - 961T^{2} \)
37 \( 1 + (2.99 + 2.99i)T + 1.36e3iT^{2} \)
41 \( 1 + 21.9iT - 1.68e3T^{2} \)
43 \( 1 + (-48.7 + 48.7i)T - 1.84e3iT^{2} \)
47 \( 1 - 70.7iT - 2.20e3T^{2} \)
53 \( 1 + (-52.8 - 52.8i)T + 2.80e3iT^{2} \)
59 \( 1 + (-61.7 + 61.7i)T - 3.48e3iT^{2} \)
61 \( 1 + (22.9 - 22.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (54.9 + 54.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 84.2T + 5.04e3T^{2} \)
73 \( 1 - 78.0iT - 5.32e3T^{2} \)
79 \( 1 - 59.2iT - 6.24e3T^{2} \)
83 \( 1 + (111. + 111. i)T + 6.88e3iT^{2} \)
89 \( 1 - 34.5iT - 7.92e3T^{2} \)
97 \( 1 + 66.0T + 9.40e3T^{2} \)
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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

−13.62336763043876207235474524062, −12.26011806851120692008505099230, −11.39424084694932397546806515051, −9.876188964744538371761660955724, −9.032345669520502802731346236953, −7.53735372528798274495005881781, −6.57694923929685135545762689857, −5.73350686299938042952713708145, −4.20212136962781781542149152946, −2.64531049466729919977750250226, 1.26281583897640058488361513646, 2.90490266856539358849561338970, 4.75430803220503427700807466663, 5.41884308672219251066738801328, 6.96592605110716669558169335628, 8.792438264515685850602752265061, 9.681521541482878144068446034959, 10.38977265355567635975452087083, 11.84361699909773155222014187457, 12.60679129628557004520363487342

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