↑

Properties

Label 2-18e2-36.31-c2-0-27
Degree $2$
Conductor $324$
Sign $0.728 - 0.684i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0560 + 1.99i)2-s + (−3.99 + 0.224i)4-s + (−0.931 + 1.61i)5-s + (9.80 − 5.65i)7-s + (−0.672 − 7.97i)8-s + (−3.27 − 1.77i)10-s + (5.08 − 2.93i)11-s + (6.96 − 12.0i)13-s + (11.8 + 19.2i)14-s + (15.8 − 1.79i)16-s − 11.0·17-s − 9.34i·19-s + (3.35 − 6.65i)20-s + (6.15 + 10.0i)22-s + (26.3 + 15.2i)23-s + ⋯
L(s)  = 1  + (0.0280 + 0.999i)2-s + (−0.998 + 0.0560i)4-s + (−0.186 + 0.322i)5-s + (1.40 − 0.808i)7-s + (−0.0840 − 0.996i)8-s + (−0.327 − 0.177i)10-s + (0.462 − 0.266i)11-s + (0.536 − 0.928i)13-s + (0.847 + 1.37i)14-s + (0.993 − 0.111i)16-s − 0.651·17-s − 0.491i·19-s + (0.167 − 0.332i)20-s + (0.279 + 0.454i)22-s + (1.14 + 0.662i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.60436 + 0.635429i\)
\(L(\frac12)\) \(\approx\) \(1.60436 + 0.635429i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0560 - 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (0.931 - 1.61i)T + (-12.5 - 21.6i)T^{2} \)
7 \( 1 + (-9.80 + 5.65i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-5.08 + 2.93i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-6.96 + 12.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 11.0T + 289T^{2} \)
19 \( 1 + 9.34iT - 361T^{2} \)
23 \( 1 + (-26.3 - 15.2i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-22.7 - 39.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (42.9 + 24.7i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 48.9T + 1.36e3T^{2} \)
41 \( 1 + (-7.44 + 12.9i)T + (-840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (4.34 - 2.51i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-71.9 + 41.5i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 53.6T + 2.80e3T^{2} \)
59 \( 1 + (85.1 + 49.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (10.2 + 17.6i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-14.9 - 8.61i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 52.6iT - 5.04e3T^{2} \)
73 \( 1 - 98.1T + 5.32e3T^{2} \)
79 \( 1 + (-2.63 + 1.52i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (88.2 - 50.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 17.0T + 7.92e3T^{2} \)
97 \( 1 + (-26.0 - 45.0i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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

−11.08524149901296196198335545143, −10.89109753270912862542628935747, −9.332745172396175609507357746449, −8.473946973738435527027674080293, −7.56200900535416064012876289831, −6.89862907853730665133943160293, −5.54599859328194246641934808543, −4.63096799728100141777011953007, −3.48389549776101379338759064972, −1.05143984768758679868936125512, 1.32796393825092792372964507763, 2.47275236567736160436918655319, 4.24206751649706363327400334016, 4.82845845021565056424999317764, 6.17581243062563675257697170236, 7.84447142148330718920566269459, 8.775551615831070632393210047354, 9.229546907425123110205162035422, 10.69704404742924701860470430947, 11.32887994090109513459886375575

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