Properties

Label 2-18e2-9.4-c7-0-24
Degree $2$
Conductor $324$
Sign $-0.766 + 0.642i$
Analytic cond. $101.212$
Root an. cond. $10.0604$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (254 + 439. i)7-s + (7.30e3 − 1.26e4i)13-s − 5.74e4·19-s + (3.90e4 + 6.76e4i)25-s + (−8.94e4 + 1.54e5i)31-s + 2.79e5·37-s + (−5.17e5 − 8.96e5i)43-s + (2.82e5 − 4.89e5i)49-s + (1.76e6 + 3.06e6i)61-s + (1.92e5 − 3.33e5i)67-s − 6.27e6·73-s + (−4.38e6 − 7.58e6i)79-s + 7.42e6·91-s + (−6.12e6 − 1.06e7i)97-s + (−4.01e6 + 6.95e6i)103-s + ⋯
L(s)  = 1  + (0.279 + 0.484i)7-s + (0.922 − 1.59i)13-s − 1.92·19-s + (0.5 + 0.866i)25-s + (−0.539 + 0.934i)31-s + 0.907·37-s + (−0.992 − 1.71i)43-s + (0.343 − 0.594i)49-s + (0.997 + 1.72i)61-s + (0.0782 − 0.135i)67-s − 1.88·73-s + (−0.999 − 1.73i)79-s + 1.03·91-s + (−0.681 − 1.17i)97-s + (−0.361 + 0.626i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(101.212\)
Root analytic conductor: \(10.0604\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :7/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7733281911\)
\(L(\frac12)\) \(\approx\) \(0.7733281911\)
\(L(\frac{9}{2})\) 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 \( 1 + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-254 - 439. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-7.30e3 + 1.26e4i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 4.10e8T^{2} \)
19 \( 1 + 5.74e4T + 8.93e8T^{2} \)
23 \( 1 + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (8.94e4 - 1.54e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 2.79e5T + 9.49e10T^{2} \)
41 \( 1 + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (5.17e5 + 8.96e5i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 1.17e12T^{2} \)
59 \( 1 + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.76e6 - 3.06e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.92e5 + 3.33e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 9.09e12T^{2} \)
73 \( 1 + 6.27e6T + 1.10e13T^{2} \)
79 \( 1 + (4.38e6 + 7.58e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 4.42e13T^{2} \)
97 \( 1 + (6.12e6 + 1.06e7i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−10.29940363016224894912420686312, −8.796756336273071924717728165691, −8.408070549404647864915456561955, −7.17782778053711466020887180993, −6.01744297277269847860418110028, −5.22272070923423882114175293939, −3.90101906820503823643514218606, −2.77367255910472664265726543727, −1.50310821646457663252130751021, −0.16543174898229926918656731752, 1.27426181892304569304764656119, 2.34901576041936390968817487368, 3.97071074778870702029558842940, 4.54766563850157560271388918364, 6.15505695424616754291157724793, 6.77035846040192467002643539159, 8.058120016828688944357972899758, 8.826140585061420586479154876158, 9.834554280364640273152137670199, 10.95119745631876226167027558609

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