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Properties

Label 2-12e2-48.11-c1-0-1
Degree $2$
Conductor $144$
Sign $-0.825 - 0.564i$
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  + (0.263 + 1.38i)2-s + (−1.86 + 0.731i)4-s + (−2.63 + 2.63i)5-s − 0.207·7-s + (−1.50 − 2.39i)8-s + (−4.35 − 2.96i)10-s + (3.66 + 3.66i)11-s + (0.255 − 0.255i)13-s + (−0.0545 − 0.287i)14-s + (2.93 − 2.72i)16-s − 0.654i·17-s + (4.46 + 4.46i)19-s + (2.97 − 6.82i)20-s + (−4.13 + 6.06i)22-s − 3.48i·23-s + ⋯
L(s)  = 1  + (0.186 + 0.982i)2-s + (−0.930 + 0.365i)4-s + (−1.17 + 1.17i)5-s − 0.0783·7-s + (−0.532 − 0.846i)8-s + (−1.37 − 0.937i)10-s + (1.10 + 1.10i)11-s + (0.0708 − 0.0708i)13-s + (−0.0145 − 0.0769i)14-s + (0.732 − 0.680i)16-s − 0.158i·17-s + (1.02 + 1.02i)19-s + (0.665 − 1.52i)20-s + (−0.880 + 1.29i)22-s − 0.727i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.825 - 0.564i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.825 - 0.564i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.825 - 0.564i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262723 + 0.850442i\)
\(L(\frac12)\) \(\approx\) \(0.262723 + 0.850442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.263 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (2.63 - 2.63i)T - 5iT^{2} \)
7 \( 1 + 0.207T + 7T^{2} \)
11 \( 1 + (-3.66 - 3.66i)T + 11iT^{2} \)
13 \( 1 + (-0.255 + 0.255i)T - 13iT^{2} \)
17 \( 1 + 0.654iT - 17T^{2} \)
19 \( 1 + (-4.46 - 4.46i)T + 19iT^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (4.33 + 4.33i)T + 29iT^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (-4.39 - 4.39i)T + 37iT^{2} \)
41 \( 1 + 0.0684T + 41T^{2} \)
43 \( 1 + (-5.65 + 5.65i)T - 43iT^{2} \)
47 \( 1 - 9.14T + 47T^{2} \)
53 \( 1 + (1.51 - 1.51i)T - 53iT^{2} \)
59 \( 1 + (2.53 + 2.53i)T + 59iT^{2} \)
61 \( 1 + (5.46 - 5.46i)T - 61iT^{2} \)
67 \( 1 + (4.77 + 4.77i)T + 67iT^{2} \)
71 \( 1 + 5.94iT - 71T^{2} \)
73 \( 1 + 6.93iT - 73T^{2} \)
79 \( 1 + 4.72iT - 79T^{2} \)
83 \( 1 + (4.32 - 4.32i)T - 83iT^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 0.925T + 97T^{2} \)
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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

−13.93711257158613464037167889787, −12.38661639841289694926603361956, −11.77268540269220008393867429733, −10.36453706529881412399772645123, −9.231984730420499862872006278112, −7.81994444576428475033281022055, −7.18600428155352425538231939579, −6.20736471539957289080660434228, −4.43790516852416569830323569940, −3.41999606282690611058903004178, 0.947942240431247221091139007976, 3.44140390952219715933913615276, 4.40401190421874285081992318574, 5.69638496590781200212928219067, 7.67922303281608873696673701054, 8.872509543714219310333028264859, 9.374436339431258801537605937290, 11.25208306528826584636285412213, 11.50725610609870841347331562205, 12.56979593147511468781615052168

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