Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} $
Sign $0.276 - 0.961i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.19 + 2.40i)2-s + (4.42 − 15.3i)4-s − 2.03·5-s + 23.1i·7-s + (22.8 + 59.7i)8-s + (6.48 − 4.88i)10-s + 4.99i·11-s + 275.·13-s + (−55.6 − 73.9i)14-s + (−216. − 136. i)16-s − 266.·17-s − 367. i·19-s + (−8.98 + 31.2i)20-s + (−12.0 − 15.9i)22-s + 628. i·23-s + ⋯
L(s)  = 1  + (−0.798 + 0.601i)2-s + (0.276 − 0.961i)4-s − 0.0812·5-s + 0.472i·7-s + (0.357 + 0.934i)8-s + (0.0648 − 0.0488i)10-s + 0.0413i·11-s + 1.63·13-s + (−0.283 − 0.377i)14-s + (−0.847 − 0.531i)16-s − 0.920·17-s − 1.01i·19-s + (−0.0224 + 0.0780i)20-s + (−0.0248 − 0.0330i)22-s + 1.18i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.276 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(324\)    =    \(2^{2} \cdot 3^{4}\)
\( \varepsilon \)  =  $0.276 - 0.961i$
motivic weight  =  \(4\)
character  :  $\chi_{324} (163, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 324,\ (\ :2),\ 0.276 - 0.961i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(1.233182962\)
\(L(\frac12)\)  \(\approx\)  \(1.233182962\)
\(L(3)\)   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 \( 1 + (3.19 - 2.40i)T \)
3 \( 1 \)
good5 \( 1 + 2.03T + 625T^{2} \)
7 \( 1 - 23.1iT - 2.40e3T^{2} \)
11 \( 1 - 4.99iT - 1.46e4T^{2} \)
13 \( 1 - 275.T + 2.85e4T^{2} \)
17 \( 1 + 266.T + 8.35e4T^{2} \)
19 \( 1 + 367. iT - 1.30e5T^{2} \)
23 \( 1 - 628. iT - 2.79e5T^{2} \)
29 \( 1 - 638.T + 7.07e5T^{2} \)
31 \( 1 + 1.37e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + 1.18e3T + 2.82e6T^{2} \)
43 \( 1 - 1.65e3iT - 3.41e6T^{2} \)
47 \( 1 - 355. iT - 4.87e6T^{2} \)
53 \( 1 - 5.29e3T + 7.89e6T^{2} \)
59 \( 1 - 6.03e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.66e3T + 1.38e7T^{2} \)
67 \( 1 - 2.20e3iT - 2.01e7T^{2} \)
71 \( 1 - 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 - 5.13e3iT - 3.89e7T^{2} \)
83 \( 1 - 7.98e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.86e3T + 6.27e7T^{2} \)
97 \( 1 - 6.81e3T + 8.85e7T^{2} \)
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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

−11.22893714920556876281782871741, −10.04631718602115565085573087431, −9.089517596368040217346853446661, −8.462868804156085667640300375430, −7.42790332690216822511625463917, −6.33764596848935896132600341619, −5.60114214865478207833597470029, −4.16613104394373279963642026974, −2.39368933062458917060028199025, −0.961804173850040452914055046593, 0.63316693150281045508901415522, 1.89623718076464104533142362404, 3.41008193373144251507533510739, 4.31120528370993691327510253583, 6.13607417352306735346619271844, 7.06927760081351958001011598361, 8.323463289061740405810012897752, 8.738094696602099956313485930791, 10.07943828194015774476483632717, 10.66834958823844094552372550952

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