Properties

Degree $2$
Conductor $324$
Sign $-0.546 - 0.837i$
Motivic weight $3$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.80 − 1.74i)5-s + (5.74 + 32.5i)7-s + (−28.2 − 10.2i)11-s + (7.59 + 6.37i)13-s + (−44.7 + 77.4i)17-s + (−29.6 − 51.3i)19-s + (17.0 − 96.7i)23-s + (−75.7 + 63.5i)25-s + (−108. + 90.6i)29-s + (4.10 − 23.2i)31-s + (84.5 + 146. i)35-s + (−114. + 197. i)37-s + (357. + 300. i)41-s + (−10.1 − 3.68i)43-s + (66.0 + 374. i)47-s + ⋯
L(s)  = 1  + (0.429 − 0.156i)5-s + (0.310 + 1.75i)7-s + (−0.773 − 0.281i)11-s + (0.162 + 0.136i)13-s + (−0.638 + 1.10i)17-s + (−0.358 − 0.620i)19-s + (0.154 − 0.876i)23-s + (−0.605 + 0.508i)25-s + (−0.692 + 0.580i)29-s + (0.0237 − 0.134i)31-s + (0.408 + 0.707i)35-s + (−0.506 + 0.877i)37-s + (1.36 + 1.14i)41-s + (−0.0358 − 0.0130i)43-s + (0.205 + 1.16i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.546 - 0.837i$
Motivic weight: \(3\)
Character: $\chi_{324} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ -0.546 - 0.837i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.287689047\)
\(L(\frac12)\) \(\approx\) \(1.287689047\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.80 + 1.74i)T + (95.7 - 80.3i)T^{2} \)
7 \( 1 + (-5.74 - 32.5i)T + (-322. + 117. i)T^{2} \)
11 \( 1 + (28.2 + 10.2i)T + (1.01e3 + 855. i)T^{2} \)
13 \( 1 + (-7.59 - 6.37i)T + (381. + 2.16e3i)T^{2} \)
17 \( 1 + (44.7 - 77.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (29.6 + 51.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-17.0 + 96.7i)T + (-1.14e4 - 4.16e3i)T^{2} \)
29 \( 1 + (108. - 90.6i)T + (4.23e3 - 2.40e4i)T^{2} \)
31 \( 1 + (-4.10 + 23.2i)T + (-2.79e4 - 1.01e4i)T^{2} \)
37 \( 1 + (114. - 197. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-357. - 300. i)T + (1.19e4 + 6.78e4i)T^{2} \)
43 \( 1 + (10.1 + 3.68i)T + (6.09e4 + 5.11e4i)T^{2} \)
47 \( 1 + (-66.0 - 374. i)T + (-9.75e4 + 3.55e4i)T^{2} \)
53 \( 1 - 202.T + 1.48e5T^{2} \)
59 \( 1 + (766. - 279. i)T + (1.57e5 - 1.32e5i)T^{2} \)
61 \( 1 + (109. + 622. i)T + (-2.13e5 + 7.76e4i)T^{2} \)
67 \( 1 + (466. + 391. i)T + (5.22e4 + 2.96e5i)T^{2} \)
71 \( 1 + (140. - 243. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (-608. - 1.05e3i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-278. + 233. i)T + (8.56e4 - 4.85e5i)T^{2} \)
83 \( 1 + (453. - 380. i)T + (9.92e4 - 5.63e5i)T^{2} \)
89 \( 1 + (-359. - 622. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-776. - 282. i)T + (6.99e5 + 5.86e5i)T^{2} \)
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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

−11.41943370107331492027739389265, −10.72062836874117394842082709030, −9.413563460716893444245494804955, −8.751556562219642960870466887651, −7.951812245580507185821434197290, −6.35712149440940284806828927417, −5.66221484877169711543777084909, −4.63444850933904569834901609885, −2.85532944775661268645937580876, −1.84844295163248005705219056143, 0.43842331590132737560858942022, 2.05103956640746957217263414326, 3.67827409614182898263001147343, 4.70454643733230536977910172742, 5.93454031239041545406494024059, 7.27633739209082246621186360719, 7.65832461741906474930009051130, 9.105493606636740897593260198167, 10.18246303835092116196316036344, 10.66015098760522336476007730597

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