Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $-0.766 + 0.642i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (3.5 − 6.06i)5-s + (7.5 − 4.33i)7-s − 7.99·8-s + (−7 − 12.1i)10-s + (7.5 − 4.33i)11-s + (−10 + 17.3i)13-s − 17.3i·14-s + (−8 + 13.8i)16-s + 8·17-s − 10.3i·19-s − 28·20-s − 17.3i·22-s + (3 + 1.73i)23-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.700 − 1.21i)5-s + (1.07 − 0.618i)7-s − 0.999·8-s + (−0.700 − 1.21i)10-s + (0.681 − 0.393i)11-s + (−0.769 + 1.33i)13-s − 1.23i·14-s + (−0.5 + 0.866i)16-s + 0.470·17-s − 0.546i·19-s − 1.40·20-s − 0.787i·22-s + (0.130 + 0.0753i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $-0.766 + 0.642i$
motivic weight  =  \(2\)
character  :  $\chi_{324} (271, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :1),\ -0.766 + 0.642i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(0.818797 - 2.24962i\)
\(L(\frac12)\)  \(\approx\)  \(0.818797 - 2.24962i\)
\(L(2)\)   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 \( 1 + (-1 + 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (-3.5 + 6.06i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-7.5 + 4.33i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.5 + 4.33i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (10 - 17.3i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 8T + 289T^{2} \)
19 \( 1 + 10.3iT - 361T^{2} \)
23 \( 1 + (-3 - 1.73i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-5 - 8.66i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (46.5 + 26.8i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + 10T + 1.36e3T^{2} \)
41 \( 1 + (25 - 43.3i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (15 - 8.66i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-75 + 43.3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 47T + 2.80e3T^{2} \)
59 \( 1 + (-30 - 17.3i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32 - 55.4i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-75 - 43.3i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 55T + 5.32e3T^{2} \)
79 \( 1 + (6 - 3.46i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-25.5 + 14.7i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 10T + 7.92e3T^{2} \)
97 \( 1 + (-12.5 - 21.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−11.32899332494845834323240062973, −10.13093106706783645956016682100, −9.241633686963044103515002561956, −8.634370133046057682848018789257, −7.08060232345992086063322935674, −5.63409472026049273549262988985, −4.80723172736558676201753911785, −4.00146740709478960464070738973, −2.02839216935656648810790280521, −1.06553234349302067555067475821, 2.27323817870885253834313826135, 3.53350842986239504052124044526, 5.12364530686031813731415793501, 5.77302663042374100094245905697, 6.93504819648759090601672873619, 7.69215276988318674419371619368, 8.736185785850588180198835842882, 9.880064674355905781753290498473, 10.81016606795631989351575136838, 11.98415199368103513079729250517

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