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Properties

Label 2-12e2-16.13-c1-0-8
Degree $2$
Conductor $144$
Sign $0.382 + 0.923i$
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 − i)2-s − 2i·4-s + (1 − i)5-s + 2i·7-s + (−2 − 2i)8-s − 2i·10-s + (−1 + i)11-s + (−1 − i)13-s + (2 + 2i)14-s − 4·16-s + 2·17-s + (3 + 3i)19-s + (−2 − 2i)20-s + 2i·22-s + 6i·23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (0.447 − 0.447i)5-s + 0.755i·7-s + (−0.707 − 0.707i)8-s − 0.632i·10-s + (−0.301 + 0.301i)11-s + (−0.277 − 0.277i)13-s + (0.534 + 0.534i)14-s − 16-s + 0.485·17-s + (0.688 + 0.688i)19-s + (−0.447 − 0.447i)20-s + 0.426i·22-s + 1.25i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30609 - 0.872707i\)
\(L(\frac12)\) \(\approx\) \(1.30609 - 0.872707i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
3 \( 1 \)
good5 \( 1 + (-1 + i)T - 5iT^{2} \)
7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + (-3 - 3i)T + 19iT^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + (3 + 3i)T + 29iT^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-3 + 3i)T - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + (-5 + 5i)T - 43iT^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 + (-3 + 3i)T - 59iT^{2} \)
61 \( 1 + (9 + 9i)T + 61iT^{2} \)
67 \( 1 + (5 + 5i)T + 67iT^{2} \)
71 \( 1 + 10iT - 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-1 - i)T + 83iT^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−12.81823279243413967006222428412, −12.09012629183502866186398697906, −11.08987876176243385929861930925, −9.818160621001463520261702823279, −9.201668185001084238726673549477, −7.54845331014245472683343796695, −5.78181063880947025148651361993, −5.21743162071581432920961001356, −3.50380959017234909227531050334, −1.89522314758274417851097890812, 2.86647072656263380760231679016, 4.34715704964063145387373736666, 5.65322604241244316825116364382, 6.81668720119359827962419983429, 7.65921835891283357581343419498, 8.992930398777413040654370520654, 10.32654918288200436091005236023, 11.37360258587122192843005460031, 12.59087709808566002578646830400, 13.48893148504962860629206560357

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