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Properties

Label 2-12e2-16.5-c1-0-4
Degree $2$
Conductor $144$
Sign $0.946 - 0.324i$
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  + (1.38 + 0.297i)2-s + (1.82 + 0.822i)4-s + (−0.595 − 0.595i)5-s + 1.64i·7-s + (2.27 + 1.68i)8-s + (−0.645 − i)10-s + (−3.36 − 3.36i)11-s + (2.64 − 2.64i)13-s + (−0.489 + 2.27i)14-s + (2.64 + 3i)16-s − 5.53·17-s + (−3.64 + 3.64i)19-s + (−0.595 − 1.57i)20-s + (−3.64 − 5.64i)22-s + 4.33i·23-s + ⋯
L(s)  = 1  + (0.977 + 0.210i)2-s + (0.911 + 0.411i)4-s + (−0.266 − 0.266i)5-s + 0.622i·7-s + (0.804 + 0.594i)8-s + (−0.204 − 0.316i)10-s + (−1.01 − 1.01i)11-s + (0.733 − 0.733i)13-s + (−0.130 + 0.608i)14-s + (0.661 + 0.750i)16-s − 1.34·17-s + (−0.836 + 0.836i)19-s + (−0.133 − 0.352i)20-s + (−0.777 − 1.20i)22-s + 0.904i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 - 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.946 - 0.324i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.946 - 0.324i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74818 + 0.291128i\)
\(L(\frac12)\) \(\approx\) \(1.74818 + 0.291128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.297i)T \)
3 \( 1 \)
good5 \( 1 + (0.595 + 0.595i)T + 5iT^{2} \)
7 \( 1 - 1.64iT - 7T^{2} \)
11 \( 1 + (3.36 + 3.36i)T + 11iT^{2} \)
13 \( 1 + (-2.64 + 2.64i)T - 13iT^{2} \)
17 \( 1 + 5.53T + 17T^{2} \)
19 \( 1 + (3.64 - 3.64i)T - 19iT^{2} \)
23 \( 1 - 4.33iT - 23T^{2} \)
29 \( 1 + (-6.12 + 6.12i)T - 29iT^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
37 \( 1 + (0.645 + 0.645i)T + 37iT^{2} \)
41 \( 1 - 7.91iT - 41T^{2} \)
43 \( 1 + (0.354 + 0.354i)T + 43iT^{2} \)
47 \( 1 - 9.10T + 47T^{2} \)
53 \( 1 + (-4.93 - 4.93i)T + 53iT^{2} \)
59 \( 1 + (4.33 + 4.33i)T + 59iT^{2} \)
61 \( 1 + (0.645 - 0.645i)T - 61iT^{2} \)
67 \( 1 + (-4 + 4i)T - 67iT^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 3.29iT - 73T^{2} \)
79 \( 1 - 9.64T + 79T^{2} \)
83 \( 1 + (-3.36 + 3.36i)T - 83iT^{2} \)
89 \( 1 - 2.38iT - 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−13.20372207991336559734156314309, −12.37116126231782002336226395027, −11.28802594992530117979489542598, −10.49669531712751581509035463511, −8.606144444004677314828634238070, −7.926153937802388552445041047937, −6.28739874221945085512495321574, −5.48423251372920476601545616352, −4.06609976759078407886084433816, −2.60477397711124635100659284735, 2.30080453982803734271724339621, 3.99089303477015943835235935838, 4.95229466002846358055187072087, 6.62860537198752915636206081847, 7.28052624755092202958837434713, 8.924113894349382568765242517915, 10.63187928500185213649095864721, 10.85332255034606171718823110941, 12.23675799924314010334687870844, 13.12526449451860419773421242130

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