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Properties

Label 2-18e2-9.2-c4-0-12
Degree $2$
Conductor $324$
Sign $0.573 + 0.819i$
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  + (14.2 − 8.20i)5-s + (23.8 − 41.2i)7-s + (95.2 + 54.9i)11-s + (17.3 + 30.0i)13-s − 208. i·17-s + 272.·19-s + (74.2 − 42.8i)23-s + (−177. + 308. i)25-s + (−552. − 318. i)29-s + (−549. − 951. i)31-s − 782. i·35-s + 219.·37-s + (2.86e3 − 1.65e3i)41-s + (−1.10e3 + 1.91e3i)43-s + (1.57e3 + 911. i)47-s + ⋯
L(s)  = 1  + (0.568 − 0.328i)5-s + (0.486 − 0.842i)7-s + (0.787 + 0.454i)11-s + (0.102 + 0.177i)13-s − 0.720i·17-s + 0.754·19-s + (0.140 − 0.0810i)23-s + (−0.284 + 0.493i)25-s + (−0.656 − 0.379i)29-s + (−0.571 − 0.990i)31-s − 0.638i·35-s + 0.159·37-s + (1.70 − 0.984i)41-s + (−0.599 + 1.03i)43-s + (0.715 + 0.412i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.573 + 0.819i$
Analytic conductor: \(33.4918\)
Root analytic conductor: \(5.78721\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :2),\ 0.573 + 0.819i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.440961770\)
\(L(\frac12)\) \(\approx\) \(2.440961770\)
\(L(3)\) 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 \( 1 \)
good5 \( 1 + (-14.2 + 8.20i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (-23.8 + 41.2i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-95.2 - 54.9i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-17.3 - 30.0i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 208. iT - 8.35e4T^{2} \)
19 \( 1 - 272.T + 1.30e5T^{2} \)
23 \( 1 + (-74.2 + 42.8i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (552. + 318. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (549. + 951. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 219.T + 1.87e6T^{2} \)
41 \( 1 + (-2.86e3 + 1.65e3i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (1.10e3 - 1.91e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.57e3 - 911. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 4.21e3iT - 7.89e6T^{2} \)
59 \( 1 + (-3.96e3 + 2.28e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (227. - 394. i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (3.48e3 + 6.04e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 7.61e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.84e3T + 2.83e7T^{2} \)
79 \( 1 + (2.36e3 - 4.10e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (3.90e3 + 2.25e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 + 5.24e3iT - 6.27e7T^{2} \)
97 \( 1 + (4.52e3 - 7.83e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−10.94908031856422234343780233349, −9.651060179073882590233685203203, −9.277284067949435830600357548886, −7.83318682882888188212852812009, −7.08530295555200668338571556130, −5.86309577964564749463182263914, −4.76332689592860738129507811082, −3.72731113420215117841767736504, −2.00113117197493232203982518786, −0.824973922732602697856840719784, 1.30786549103588220869015308933, 2.55776556415717188096934029414, 3.89041187326583531545168310545, 5.39542842398358729205602461247, 6.06859716554324315294864149319, 7.23633421910125936091345913999, 8.470307642754955336101559793060, 9.159396638741291122742823281798, 10.20583915115801974067794491933, 11.16098147645852531315756258176

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