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Properties

Label 2-12e2-48.11-c3-0-1
Degree $2$
Conductor $144$
Sign $0.271 - 0.962i$
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  + (−1.73 − 2.23i)2-s + (−1.96 + 7.75i)4-s + (3.22 − 3.22i)5-s − 13.1·7-s + (20.7 − 9.07i)8-s + (−12.7 − 1.59i)10-s + (−3.39 − 3.39i)11-s + (−54.1 + 54.1i)13-s + (22.7 + 29.2i)14-s + (−56.2 − 30.5i)16-s + 70.7i·17-s + (32.5 + 32.5i)19-s + (18.6 + 31.3i)20-s + (−1.68 + 13.4i)22-s − 16.4i·23-s + ⋯
L(s)  = 1  + (−0.613 − 0.789i)2-s + (−0.246 + 0.969i)4-s + (0.288 − 0.288i)5-s − 0.707·7-s + (0.916 − 0.400i)8-s + (−0.404 − 0.0505i)10-s + (−0.0929 − 0.0929i)11-s + (−1.15 + 1.15i)13-s + (0.434 + 0.558i)14-s + (−0.878 − 0.476i)16-s + 1.00i·17-s + (0.393 + 0.393i)19-s + (0.208 + 0.350i)20-s + (−0.0163 + 0.130i)22-s − 0.148i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(8.49627\)
Root analytic conductor: \(2.91483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :3/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.467643 + 0.354160i\)
\(L(\frac12)\) \(\approx\) \(0.467643 + 0.354160i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73 + 2.23i)T \)
3 \( 1 \)
good5 \( 1 + (-3.22 + 3.22i)T - 125iT^{2} \)
7 \( 1 + 13.1T + 343T^{2} \)
11 \( 1 + (3.39 + 3.39i)T + 1.33e3iT^{2} \)
13 \( 1 + (54.1 - 54.1i)T - 2.19e3iT^{2} \)
17 \( 1 - 70.7iT - 4.91e3T^{2} \)
19 \( 1 + (-32.5 - 32.5i)T + 6.85e3iT^{2} \)
23 \( 1 + 16.4iT - 1.21e4T^{2} \)
29 \( 1 + (-28.0 - 28.0i)T + 2.43e4iT^{2} \)
31 \( 1 - 174. iT - 2.97e4T^{2} \)
37 \( 1 + (-116. - 116. i)T + 5.06e4iT^{2} \)
41 \( 1 + 19.6T + 6.89e4T^{2} \)
43 \( 1 + (-94.2 + 94.2i)T - 7.95e4iT^{2} \)
47 \( 1 + 372.T + 1.03e5T^{2} \)
53 \( 1 + (162. - 162. i)T - 1.48e5iT^{2} \)
59 \( 1 + (610. + 610. i)T + 2.05e5iT^{2} \)
61 \( 1 + (531. - 531. i)T - 2.26e5iT^{2} \)
67 \( 1 + (-562. - 562. i)T + 3.00e5iT^{2} \)
71 \( 1 + 1.16e3iT - 3.57e5T^{2} \)
73 \( 1 + 308. iT - 3.89e5T^{2} \)
79 \( 1 + 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 + (469. - 469. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.53e3T + 7.04e5T^{2} \)
97 \( 1 + 139.T + 9.12e5T^{2} \)
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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

−12.60816978569550986134075453608, −11.89392777679204154382673719345, −10.67089531273002326351267840926, −9.703141823010428183047313750941, −9.032536373372756139377990572535, −7.74315544881804001490329770332, −6.52419169671319011991945837805, −4.74168643113894418763958522319, −3.27624409602647547006815393673, −1.70668190053497230513030487941, 0.34505356188162308434438061434, 2.69061785434575330746905814930, 4.86612100739378905079325879454, 6.01886699679801626626076261604, 7.12761859716433253884224429456, 8.019379006192421307506577764526, 9.541270950935529811833910929325, 9.914077830610764552473073899881, 11.14315111600815352307441573121, 12.57157594592346343826608849062

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