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Properties

Label 2-648-9.4-c3-0-11
Degree $2$
Conductor $648$
Sign $0.766 - 0.642i$
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  + (6.61 − 11.4i)5-s + (2.61 + 4.53i)7-s + (−0.5 − 0.866i)11-s + (−42.4 + 73.5i)13-s + 40.9·17-s + 57.5·19-s + (−57.4 + 99.5i)23-s + (−25.0 − 43.4i)25-s + (101. + 175. i)29-s + (−137. + 237. i)31-s + 69.2·35-s + 242.·37-s + (164. − 284. i)41-s + (−140. − 243. i)43-s + (−11.9 − 20.6i)47-s + ⋯
L(s)  = 1  + (0.591 − 1.02i)5-s + (0.141 + 0.244i)7-s + (−0.0137 − 0.0237i)11-s + (−0.906 + 1.56i)13-s + 0.584·17-s + 0.694·19-s + (−0.520 + 0.902i)23-s + (−0.200 − 0.347i)25-s + (0.648 + 1.12i)29-s + (−0.794 + 1.37i)31-s + 0.334·35-s + 1.07·37-s + (0.625 − 1.08i)41-s + (−0.498 − 0.863i)43-s + (−0.0370 − 0.0641i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.766 - 0.642i$
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: \((2,\ 648,\ (\ :3/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.032180040\)
\(L(\frac12)\) \(\approx\) \(2.032180040\)
\(L(\frac{5}{2})\) 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 \( 1 + (-6.61 + 11.4i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-2.61 - 4.53i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (42.4 - 73.5i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 - 40.9T + 4.91e3T^{2} \)
19 \( 1 - 57.5T + 6.85e3T^{2} \)
23 \( 1 + (57.4 - 99.5i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-101. - 175. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (137. - 237. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 242.T + 5.06e4T^{2} \)
41 \( 1 + (-164. + 284. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (140. + 243. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (11.9 + 20.6i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 300.T + 1.48e5T^{2} \)
59 \( 1 + (376. - 652. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (247. + 429. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-204. + 355. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 1.11e3T + 3.57e5T^{2} \)
73 \( 1 + 287T + 3.89e5T^{2} \)
79 \( 1 + (-615. - 1.06e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-471. - 816. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 190.T + 7.04e5T^{2} \)
97 \( 1 + (-153. - 265. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−10.02487673228873351488604047850, −9.274111427356982056228245737250, −8.805540876144127905145354376991, −7.59928879975051299937818768954, −6.73092954364027762103519493224, −5.43761231980465087751835348140, −4.97838245186175863956073728090, −3.72626738756349847468862596662, −2.17405776056314711634556591451, −1.19038727003259408549529419799, 0.63385827379449684997981473247, 2.38301187325602658923676377101, 3.10633743045110584389863386401, 4.51145503611446857923976954851, 5.69672577813421377428235598224, 6.36386691371157185016865411323, 7.60958571157254963628394560606, 7.959494189072008541647300857465, 9.565206729793050200125745135048, 10.04575963794254350695003838908

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