Properties

Degree $2$
Conductor $324$
Sign $-0.477 + 0.878i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.0 − 3.64i)5-s + (−2.90 − 16.4i)7-s + (−3.53 − 1.28i)11-s + (−55.0 − 46.1i)13-s + (14.0 − 24.3i)17-s + (4.34 + 7.52i)19-s + (−4.21 + 23.9i)23-s + (−8.84 + 7.42i)25-s + (−183. + 153. i)29-s + (45.3 − 257. i)31-s + (−89.2 − 154. i)35-s + (50.0 − 86.6i)37-s + (−177. − 149. i)41-s + (220. + 80.1i)43-s + (−98.9 − 561. i)47-s + ⋯
L(s)  = 1  + (0.895 − 0.325i)5-s + (−0.157 − 0.890i)7-s + (−0.0969 − 0.0352i)11-s + (−1.17 − 0.985i)13-s + (0.200 − 0.347i)17-s + (0.0524 + 0.0908i)19-s + (−0.0382 + 0.216i)23-s + (−0.0707 + 0.0593i)25-s + (−1.17 + 0.983i)29-s + (0.262 − 1.49i)31-s + (−0.430 − 0.746i)35-s + (0.222 − 0.385i)37-s + (−0.676 − 0.567i)41-s + (0.781 + 0.284i)43-s + (−0.307 − 1.74i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.477 + 0.878i$
Motivic weight: \(3\)
Character: $\chi_{324} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.477 + 0.878i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.454125968\)
\(L(\frac12)\) \(\approx\) \(1.454125968\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-10.0 + 3.64i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (2.90 + 16.4i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (3.53 + 1.28i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (55.0 + 46.1i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (-14.0 + 24.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-4.34 - 7.52i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (4.21 - 23.9i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (183. - 153. i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-45.3 + 257. i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (-50.0 + 86.6i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (177. + 149. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (-220. - 80.1i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (98.9 + 561. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 + 368.T + 1.48e5T^{2} \)
59 \( 1 + (-485. + 176. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (106. + 605. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (264. + 221. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (496. - 860. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (180. + 312. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-127. + 107. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (-718. + 602. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-404. - 700. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.72e3 - 627. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−10.66590200573752564788938680405, −9.923685253935838891863921061100, −9.263327800049530553367004896054, −7.86193974002579877113312802747, −7.12527654274071976755195636479, −5.78854637840905455600992560400, −4.98494482236257972214459664097, −3.54907773376351771785457050869, −2.09068306885166226243582195871, −0.49209724978456709173620084685, 1.86810995677106013727108437565, 2.82703265220755932420869619049, 4.54000011581618714968947979250, 5.69029106756110078722374927173, 6.49628758824857252502189977270, 7.61536631046582023876556591456, 8.913301435057783794362221979400, 9.607302678348329363882825827331, 10.38826129545416990642862512347, 11.61106397356347853727730882515

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