↑

Properties

Label 2-18e2-324.263-c1-0-13
Degree $2$
Conductor $324$
Sign $-0.989 - 0.145i$
Analytic cond. $2.58715$
Root an. cond. $1.60846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.08 + 0.911i)2-s + (−1.15 + 1.29i)3-s + (0.338 + 1.97i)4-s + (−0.676 + 1.34i)5-s + (−2.42 + 0.346i)6-s + (−1.90 + 0.569i)7-s + (−1.42 + 2.44i)8-s + (−0.339 − 2.98i)9-s + (−1.96 + 0.840i)10-s + (0.0271 − 0.465i)11-s + (−2.93 − 1.83i)12-s + (2.08 − 2.80i)13-s + (−2.57 − 1.11i)14-s + (−0.960 − 2.42i)15-s + (−3.77 + 1.33i)16-s + (1.49 + 1.78i)17-s + ⋯
L(s)  = 1  + (0.764 + 0.644i)2-s + (−0.665 + 0.746i)3-s + (0.169 + 0.985i)4-s + (−0.302 + 0.602i)5-s + (−0.989 + 0.141i)6-s + (−0.718 + 0.215i)7-s + (−0.505 + 0.862i)8-s + (−0.113 − 0.993i)9-s + (−0.619 + 0.265i)10-s + (0.00818 − 0.140i)11-s + (−0.848 − 0.529i)12-s + (0.578 − 0.776i)13-s + (−0.688 − 0.298i)14-s + (−0.248 − 0.627i)15-s + (−0.942 + 0.333i)16-s + (0.362 + 0.432i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.145i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.989 - 0.145i$
Analytic conductor: \(2.58715\)
Root analytic conductor: \(1.60846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1/2),\ -0.989 - 0.145i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0894094 + 1.22419i\)
\(L(\frac12)\) \(\approx\) \(0.0894094 + 1.22419i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 - 0.911i)T \)
3 \( 1 + (1.15 - 1.29i)T \)
good5 \( 1 + (0.676 - 1.34i)T + (-2.98 - 4.01i)T^{2} \)
7 \( 1 + (1.90 - 0.569i)T + (5.84 - 3.84i)T^{2} \)
11 \( 1 + (-0.0271 + 0.465i)T + (-10.9 - 1.27i)T^{2} \)
13 \( 1 + (-2.08 + 2.80i)T + (-3.72 - 12.4i)T^{2} \)
17 \( 1 + (-1.49 - 1.78i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (1.70 - 2.03i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (1.76 - 5.87i)T + (-19.2 - 12.6i)T^{2} \)
29 \( 1 + (-4.45 - 1.92i)T + (19.9 + 21.0i)T^{2} \)
31 \( 1 + (2.49 - 2.35i)T + (1.80 - 30.9i)T^{2} \)
37 \( 1 + (0.895 + 5.07i)T + (-34.7 + 12.6i)T^{2} \)
41 \( 1 + (1.31 - 11.2i)T + (-39.8 - 9.45i)T^{2} \)
43 \( 1 + (-4.58 + 6.97i)T + (-17.0 - 39.4i)T^{2} \)
47 \( 1 + (-4.87 + 5.16i)T + (-2.73 - 46.9i)T^{2} \)
53 \( 1 + (-1.98 + 1.14i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.428 - 7.36i)T + (-58.6 + 6.84i)T^{2} \)
61 \( 1 + (3.25 + 0.771i)T + (54.5 + 27.3i)T^{2} \)
67 \( 1 + (-9.27 + 4.00i)T + (45.9 - 48.7i)T^{2} \)
71 \( 1 + (-12.9 - 4.70i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-10.3 + 3.75i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (1.24 + 10.6i)T + (-76.8 + 18.2i)T^{2} \)
83 \( 1 + (-6.60 + 0.771i)T + (80.7 - 19.1i)T^{2} \)
89 \( 1 + (1.79 + 4.92i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (11.6 - 5.86i)T + (57.9 - 77.8i)T^{2} \)
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

−12.16836814452724081180235382606, −11.16423583296046679108155570368, −10.43324121712228266591370927769, −9.253532052322376282173824070366, −8.116662901649694253740558377946, −6.92455887252804359055893977375, −6.04082839293106418814266535042, −5.31120216761276357676218673076, −3.83314493626604899973441800530, −3.20467195103342559347548215319, 0.75420232871298985893152862495, 2.41632891971637469704614122639, 4.07311236837263387889356113211, 5.02484896588307676878874834119, 6.26950287435583611326699015880, 6.85681037249988463657892630737, 8.334523873612105888253133072663, 9.526576093224307626673423903576, 10.59915126627457880667757526984, 11.37003831891203938877923254890

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