Properties

Label 2-18e2-12.11-c3-0-49
Degree $2$
Conductor $324$
Sign $0.748 + 0.663i$
Analytic cond. $19.1166$
Root an. cond. $4.37225$
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  + (1.16 + 2.57i)2-s + (−5.30 + 5.98i)4-s − 16.7i·5-s + 19.3i·7-s + (−21.5 − 6.74i)8-s + (43.1 − 19.4i)10-s − 4.88·11-s − 12.0·13-s + (−49.7 + 22.3i)14-s + (−7.64 − 63.5i)16-s − 71.2i·17-s − 68.3i·19-s + (100. + 88.8i)20-s + (−5.66 − 12.5i)22-s + 136.·23-s + ⋯
L(s)  = 1  + (0.410 + 0.912i)2-s + (−0.663 + 0.748i)4-s − 1.49i·5-s + 1.04i·7-s + (−0.954 − 0.298i)8-s + (1.36 − 0.613i)10-s − 0.133·11-s − 0.257·13-s + (−0.950 + 0.427i)14-s + (−0.119 − 0.992i)16-s − 1.01i·17-s − 0.824i·19-s + (1.11 + 0.993i)20-s + (−0.0548 − 0.122i)22-s + 1.23·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.748 + 0.663i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.748 + 0.663i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.439709181\)
\(L(\frac12)\) \(\approx\) \(1.439709181\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 - 2.57i)T \)
3 \( 1 \)
good5 \( 1 + 16.7iT - 125T^{2} \)
7 \( 1 - 19.3iT - 343T^{2} \)
11 \( 1 + 4.88T + 1.33e3T^{2} \)
13 \( 1 + 12.0T + 2.19e3T^{2} \)
17 \( 1 + 71.2iT - 4.91e3T^{2} \)
19 \( 1 + 68.3iT - 6.85e3T^{2} \)
23 \( 1 - 136.T + 1.21e4T^{2} \)
29 \( 1 + 219. iT - 2.43e4T^{2} \)
31 \( 1 + 329. iT - 2.97e4T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + 34.1iT - 6.89e4T^{2} \)
43 \( 1 - 0.644iT - 7.95e4T^{2} \)
47 \( 1 + 186.T + 1.03e5T^{2} \)
53 \( 1 - 266. iT - 1.48e5T^{2} \)
59 \( 1 - 208.T + 2.05e5T^{2} \)
61 \( 1 + 1.60T + 2.26e5T^{2} \)
67 \( 1 - 428. iT - 3.00e5T^{2} \)
71 \( 1 + 386.T + 3.57e5T^{2} \)
73 \( 1 + 776.T + 3.89e5T^{2} \)
79 \( 1 + 79.0iT - 4.93e5T^{2} \)
83 \( 1 - 925.T + 5.71e5T^{2} \)
89 \( 1 + 1.04e3iT - 7.04e5T^{2} \)
97 \( 1 + 1.46e3T + 9.12e5T^{2} \)
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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

−11.55923398495198197209900742742, −9.551536883839080407005288719525, −9.067590895369872296025284266734, −8.272782928453937003850199007891, −7.25643978198557532614605766446, −5.89875811142801455837666278396, −5.14107234501113055909023233094, −4.38260656841917081588744603584, −2.63821028474659719918125527594, −0.46816308184580926995506014128, 1.49818136120132220082503357197, 3.04526849228951043310862563163, 3.73248508771212670413528387006, 5.10543196534326905519255262750, 6.47395629425784733738228923519, 7.23095941266747075773742951237, 8.605825103223543533395234084187, 9.963218970228532172980728616898, 10.66120603908259081499333725331, 10.90295961398094489231278624141

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