↑

Properties

Label 2-18e2-9.4-c3-0-7
Degree $2$
Conductor $324$
Sign $0.939 + 0.342i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 7.79i)5-s + (0.5 + 0.866i)7-s + (31.5 + 54.5i)11-s + (14 − 24.2i)13-s − 72·17-s + 98·19-s + (63 − 109. i)23-s + (22 + 38.1i)25-s + (−63 − 109. i)29-s + (129.5 − 224. i)31-s + 9·35-s + 386·37-s + (−225 + 389. i)41-s + (17 + 29.4i)43-s + (−27 − 46.7i)47-s + ⋯
L(s)  = 1  + (0.402 − 0.697i)5-s + (0.0269 + 0.0467i)7-s + (0.863 + 1.49i)11-s + (0.298 − 0.517i)13-s − 1.02·17-s + 1.18·19-s + (0.571 − 0.989i)23-s + (0.175 + 0.304i)25-s + (−0.403 − 0.698i)29-s + (0.750 − 1.29i)31-s + 0.0434·35-s + 1.71·37-s + (−0.857 + 1.48i)41-s + (0.0602 + 0.104i)43-s + (−0.0837 − 0.145i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(19.1166\)
Root analytic conductor: \(4.37225\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :3/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.147276765\)
\(L(\frac12)\) \(\approx\) \(2.147276765\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.5 + 7.79i)T + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-31.5 - 54.5i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (-14 + 24.2i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 72T + 4.91e3T^{2} \)
19 \( 1 - 98T + 6.85e3T^{2} \)
23 \( 1 + (-63 + 109. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (63 + 109. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-129.5 + 224. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 - 386T + 5.06e4T^{2} \)
41 \( 1 + (225 - 389. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-17 - 29.4i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (27 + 46.7i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 - 693T + 1.48e5T^{2} \)
59 \( 1 + (-90 + 155. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-140 - 242. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-293 + 507. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 504T + 3.57e5T^{2} \)
73 \( 1 - 161T + 3.89e5T^{2} \)
79 \( 1 + (220 + 381. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-499.5 - 865. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 882T + 7.04e5T^{2} \)
97 \( 1 + (-360.5 - 624. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.27490921265456639126882499227, −9.914226892957083312216351734028, −9.414098107914900671332442148881, −8.394291053613700165521075090540, −7.25338042637745938199815086003, −6.24751173837841826179422627670, −5.01610469932105822388424082171, −4.15040617808194369854264522000, −2.39685750186238151734701212805, −1.01734501259156112813752983396, 1.14882764247013419221023359712, 2.82544755386384763771918236647, 3.88251553528622753647804949070, 5.43146876738338326187065693893, 6.42038369389618238693571707480, 7.18439776461159490621484937667, 8.624315827150434228927878409307, 9.241152309934714861075373218607, 10.44558533543398950030477442776, 11.25071340806086073110414500933

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