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Properties

Label 2-18e2-4.3-c4-0-66
Degree $2$
Conductor $324$
Sign $0.768 + 0.640i$
Analytic cond. $33.4918$
Root an. cond. $5.78721$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.76 + 1.36i)2-s + (12.2 + 10.2i)4-s − 28.6·5-s − 25.6i·7-s + (32.3 + 55.2i)8-s + (−107. − 38.9i)10-s − 108. i·11-s − 88.4·13-s + (34.9 − 96.4i)14-s + (46.2 + 251. i)16-s + 504.·17-s − 191. i·19-s + (−351. − 292. i)20-s + (147. − 408. i)22-s − 960. i·23-s + ⋯
L(s)  = 1  + (0.940 + 0.340i)2-s + (0.768 + 0.640i)4-s − 1.14·5-s − 0.523i·7-s + (0.504 + 0.863i)8-s + (−1.07 − 0.389i)10-s − 0.896i·11-s − 0.523·13-s + (0.178 − 0.492i)14-s + (0.180 + 0.983i)16-s + 1.74·17-s − 0.530i·19-s + (−0.879 − 0.732i)20-s + (0.305 − 0.843i)22-s − 1.81i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.768 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.768 + 0.640i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ 0.768 + 0.640i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.645997359\)
\(L(\frac12)\) \(\approx\) \(2.645997359\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.76 - 1.36i)T \)
3 \( 1 \)
good5 \( 1 + 28.6T + 625T^{2} \)
7 \( 1 + 25.6iT - 2.40e3T^{2} \)
11 \( 1 + 108. iT - 1.46e4T^{2} \)
13 \( 1 + 88.4T + 2.85e4T^{2} \)
17 \( 1 - 504.T + 8.35e4T^{2} \)
19 \( 1 + 191. iT - 1.30e5T^{2} \)
23 \( 1 + 960. iT - 2.79e5T^{2} \)
29 \( 1 - 793.T + 7.07e5T^{2} \)
31 \( 1 + 329. iT - 9.23e5T^{2} \)
37 \( 1 - 209.T + 1.87e6T^{2} \)
41 \( 1 - 1.05e3T + 2.82e6T^{2} \)
43 \( 1 + 3.33e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.12e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.13e3T + 7.89e6T^{2} \)
59 \( 1 + 4.66e3iT - 1.21e7T^{2} \)
61 \( 1 + 5.59e3T + 1.38e7T^{2} \)
67 \( 1 - 7.07e3iT - 2.01e7T^{2} \)
71 \( 1 + 4.43e3iT - 2.54e7T^{2} \)
73 \( 1 + 1.95e3T + 2.83e7T^{2} \)
79 \( 1 - 1.75e3iT - 3.89e7T^{2} \)
83 \( 1 + 3.02e3iT - 4.74e7T^{2} \)
89 \( 1 + 559.T + 6.27e7T^{2} \)
97 \( 1 + 2.20e3T + 8.85e7T^{2} \)
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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

−11.08971340045577757737481328031, −10.27994281803804549068269519823, −8.581476507771204438609011194786, −7.80506928156578621618173680930, −7.04536907799641527158661841975, −5.88315519507294744378812759597, −4.69176234053672487035174892913, −3.78582875022657800015207444335, −2.79990066867054097563096493562, −0.62116415158017491047543175205, 1.34447517682553433654520760223, 2.90239638655948038851289237189, 3.88439989121975718555900387638, 4.94992758820066454871251654533, 5.91925407822436312880097368258, 7.34307527722601314485123884905, 7.84245395274929963331245471335, 9.495955964515605601740149430559, 10.27516692096316214401615939025, 11.48525992198491922137208053731

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