Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.986 + 0.163i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.328 − 3.98i)2-s + (−15.7 + 2.61i)4-s + 5.66·5-s + 52.1i·7-s + (15.6 + 62.0i)8-s + (−1.85 − 22.5i)10-s + 106. i·11-s − 122.·13-s + (207. − 17.1i)14-s + (242. − 82.6i)16-s − 122.·17-s − 593. i·19-s + (−89.3 + 14.8i)20-s + (425. − 35.0i)22-s − 546. i·23-s + ⋯
L(s)  = 1  + (−0.0820 − 0.996i)2-s + (−0.986 + 0.163i)4-s + 0.226·5-s + 1.06i·7-s + (0.244 + 0.969i)8-s + (−0.0185 − 0.225i)10-s + 0.881i·11-s − 0.722·13-s + (1.06 − 0.0873i)14-s + (0.946 − 0.322i)16-s − 0.424·17-s − 1.64i·19-s + (−0.223 + 0.0370i)20-s + (0.878 − 0.0723i)22-s − 1.03i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.986 + 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.986 + 0.163i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ -0.986 + 0.163i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(0.6759784689\)
\(L(\frac12)\)  \(\approx\)  \(0.6759784689\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.328 + 3.98i)T \)
3 \( 1 \)
good5 \( 1 - 5.66T + 625T^{2} \)
7 \( 1 - 52.1iT - 2.40e3T^{2} \)
11 \( 1 - 106. iT - 1.46e4T^{2} \)
13 \( 1 + 122.T + 2.85e4T^{2} \)
17 \( 1 + 122.T + 8.35e4T^{2} \)
19 \( 1 + 593. iT - 1.30e5T^{2} \)
23 \( 1 + 546. iT - 2.79e5T^{2} \)
29 \( 1 - 735.T + 7.07e5T^{2} \)
31 \( 1 + 585. iT - 9.23e5T^{2} \)
37 \( 1 - 2.28e3T + 1.87e6T^{2} \)
41 \( 1 + 2.86e3T + 2.82e6T^{2} \)
43 \( 1 + 2.24e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.05e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.75e3T + 7.89e6T^{2} \)
59 \( 1 - 2.15e3iT - 1.21e7T^{2} \)
61 \( 1 + 66.3T + 1.38e7T^{2} \)
67 \( 1 + 4.10e3iT - 2.01e7T^{2} \)
71 \( 1 + 5.03e3iT - 2.54e7T^{2} \)
73 \( 1 - 2.70e3T + 2.83e7T^{2} \)
79 \( 1 + 1.38e3iT - 3.89e7T^{2} \)
83 \( 1 - 3.00e3iT - 4.74e7T^{2} \)
89 \( 1 - 3.18e3T + 6.27e7T^{2} \)
97 \( 1 + 4.81e3T + 8.85e7T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−10.53420863784485197764382547873, −9.593943057819786894632105189393, −9.007488325073510321654682303725, −7.948783019530554569587936820513, −6.58335576206291826531812817233, −5.19085116729961343819719838765, −4.42402239513752708900581067359, −2.72873476394137938549147938877, −2.06231545625231111476272981836, −0.22219667904574917224971608095, 1.24632142115537407355125626732, 3.45105885980072437710927866408, 4.51200718653855275335570624870, 5.70657502982566750465668517539, 6.57888947694723732245359108084, 7.66508761712414799293102608609, 8.271462234515167728531387525474, 9.616795907349793366794501363615, 10.14612736216379084381935739271, 11.30866160887086778508915376274

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