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Properties

Label 2-648-9.4-c3-0-34
Degree $2$
Conductor $648$
Sign $-0.939 - 0.342i$
Analytic cond. $38.2332$
Root an. cond. $6.18330$
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  + (7.70 − 13.3i)5-s + (−11.9 − 20.6i)7-s + (−7.12 − 12.3i)11-s + (6.91 − 11.9i)13-s − 80.5·17-s − 144.·19-s + (70.5 − 122. i)23-s + (−56.3 − 97.5i)25-s + (125. + 217. i)29-s + (8.33 − 14.4i)31-s − 367.·35-s + 305.·37-s + (−214. + 371. i)41-s + (90.8 + 157. i)43-s + (39.7 + 68.7i)47-s + ⋯
L(s)  = 1  + (0.689 − 1.19i)5-s + (−0.643 − 1.11i)7-s + (−0.195 − 0.338i)11-s + (0.147 − 0.255i)13-s − 1.14·17-s − 1.74·19-s + (0.639 − 1.10i)23-s + (−0.450 − 0.780i)25-s + (0.804 + 1.39i)29-s + (0.0482 − 0.0836i)31-s − 1.77·35-s + 1.35·37-s + (−0.817 + 1.41i)41-s + (0.322 + 0.557i)43-s + (0.123 + 0.213i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(38.2332\)
Root analytic conductor: \(6.18330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :3/2),\ -0.939 - 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.8205371939\)
\(L(\frac12)\) \(\approx\) \(0.8205371939\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-7.70 + 13.3i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (11.9 + 20.6i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (7.12 + 12.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-6.91 + 11.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 80.5T + 4.91e3T^{2} \)
19 \( 1 + 144.T + 6.85e3T^{2} \)
23 \( 1 + (-70.5 + 122. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-125. - 217. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-8.33 + 14.4i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 305.T + 5.06e4T^{2} \)
41 \( 1 + (214. - 371. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-90.8 - 157. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-39.7 - 68.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 663.T + 1.48e5T^{2} \)
59 \( 1 + (110. - 190. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-236. - 409. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-323. + 560. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 14.4T + 3.57e5T^{2} \)
73 \( 1 - 776.T + 3.89e5T^{2} \)
79 \( 1 + (-128. - 223. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (642. + 1.11e3i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 156.T + 7.04e5T^{2} \)
97 \( 1 + (580. + 1.00e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−9.638597409699459699375678664763, −8.792801568700449692288718151867, −8.172444877479188221869170683601, −6.75233890575325223294881566161, −6.22567769541823806087956936595, −4.85410802747072247727010913998, −4.26895116756944567091658777333, −2.79386925710880524474666531158, −1.32494714686276510785778733033, −0.22651305048014991350625963652, 2.15910359504680760512463596943, 2.63776854324253475319595232904, 4.01442633695128299123600373140, 5.38603532394268862074831551448, 6.48306748116878344935155062519, 6.62092583504775877407188485696, 8.076333629331750164024578289322, 9.099451923015545927916359968198, 9.719301227686698336912317260655, 10.65839642263434497894493414845

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