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Properties

Label 2-18e2-9.4-c3-0-1
Degree $2$
Conductor $324$
Sign $-0.173 - 0.984i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.5 − 14.7i)7-s + (−44.5 + 77.0i)13-s + 107·19-s + (62.5 + 108. i)25-s + (−154 + 266. i)31-s − 433·37-s + (260 + 450. i)43-s + (27 − 46.7i)49-s + (450.5 + 780. i)61-s + (−503.5 + 872. i)67-s − 271·73-s + (−251.5 − 435. i)79-s + 1.51e3·91-s + (−926.5 − 1.60e3i)97-s + (9.5 − 16.4i)103-s + ⋯
L(s)  = 1  + (−0.458 − 0.794i)7-s + (−0.949 + 1.64i)13-s + 1.29·19-s + (0.5 + 0.866i)25-s + (−0.892 + 1.54i)31-s − 1.92·37-s + (0.922 + 1.59i)43-s + (0.0787 − 0.136i)49-s + (0.945 + 1.63i)61-s + (−0.918 + 1.59i)67-s − 0.434·73-s + (−0.358 − 0.620i)79-s + 1.74·91-s + (−0.969 − 1.67i)97-s + (0.00908 − 0.0157i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
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: \((2,\ 324,\ (\ :3/2),\ -0.173 - 0.984i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.044747439\)
\(L(\frac12)\) \(\approx\) \(1.044747439\)
\(L(\frac{5}{2})\) 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 \( 1 + (-62.5 - 108. i)T^{2} \)
7 \( 1 + (8.5 + 14.7i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + (44.5 - 77.0i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + 4.91e3T^{2} \)
19 \( 1 - 107T + 6.85e3T^{2} \)
23 \( 1 + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + 433T + 5.06e4T^{2} \)
41 \( 1 + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-260 - 450. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-450.5 - 780. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (503.5 - 872. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 + 271T + 3.89e5T^{2} \)
79 \( 1 + (251.5 + 435. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + 7.04e5T^{2} \)
97 \( 1 + (926.5 + 1.60e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.49380393827533142355984799335, −10.44787895345491171515221873567, −9.576388470309955485764095959695, −8.812143734218128454249921915932, −7.18609054441951655923459920399, −7.01958430435382091365329986772, −5.42190152205182772722049638806, −4.31691669560371039256019514301, −3.12972756338506553802889723536, −1.45428318178761206728967840321, 0.38151929726089190924900171525, 2.39704149985424468385239901056, 3.44595979829924927111067781997, 5.12903352169656611318515130536, 5.80002858783243465787826093162, 7.15276909921579537655604378267, 8.036583517604086068262059213962, 9.144392188729630185390396760906, 9.955576646757504687975032636207, 10.83481079229787325935131551386

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