↑

Properties

Label 2-804-201.38-c1-0-2
Degree $2$
Conductor $804$
Sign $-0.00870 - 0.999i$
Analytic cond. $6.41997$
Root an. cond. $2.53376$
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  + (−1.53 − 0.798i)3-s − 3.20·5-s + (2.52 − 1.45i)7-s + (1.72 + 2.45i)9-s + (−0.987 − 1.70i)11-s + (−4.07 − 2.35i)13-s + (4.92 + 2.55i)15-s + (−2.01 − 1.16i)17-s + (−3.36 + 5.83i)19-s + (−5.05 + 0.225i)21-s + (4.96 + 2.86i)23-s + 5.26·25-s + (−0.692 − 5.14i)27-s + (7.75 − 4.48i)29-s + (−0.441 + 0.254i)31-s + ⋯
L(s)  = 1  + (−0.887 − 0.460i)3-s − 1.43·5-s + (0.955 − 0.551i)7-s + (0.575 + 0.818i)9-s + (−0.297 − 0.515i)11-s + (−1.13 − 0.652i)13-s + (1.27 + 0.660i)15-s + (−0.488 − 0.281i)17-s + (−0.772 + 1.33i)19-s + (−1.10 + 0.0491i)21-s + (1.03 + 0.597i)23-s + 1.05·25-s + (−0.133 − 0.991i)27-s + (1.44 − 0.831i)29-s + (−0.0792 + 0.0457i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00870 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00870 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.00870 - 0.999i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.00870 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248107 + 0.250275i\)
\(L(\frac12)\) \(\approx\) \(0.248107 + 0.250275i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.53 + 0.798i)T \)
67 \( 1 + (2.26 + 7.86i)T \)
good5 \( 1 + 3.20T + 5T^{2} \)
7 \( 1 + (-2.52 + 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.987 + 1.70i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.07 + 2.35i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.01 + 1.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.36 - 5.83i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.96 - 2.86i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.75 + 4.48i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.441 - 0.254i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (5.77 - 10.0i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.44 - 7.69i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.8iT - 43T^{2} \)
47 \( 1 + (10.8 - 6.26i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.44T + 53T^{2} \)
59 \( 1 - 3.19iT - 59T^{2} \)
61 \( 1 + (-4.86 - 2.80i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (7.69 - 4.44i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.977 + 1.69i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.79 + 5.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.1 + 6.43i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.87iT - 89T^{2} \)
97 \( 1 + (5.04 + 2.91i)T + (48.5 + 84.0i)T^{2} \)
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

−10.71530123774155433370315643282, −9.933162543029672082885300683800, −8.123537103381983351691989119158, −8.050222711762511432333239315796, −7.17619427674470210189792279191, −6.17899206147608741832731662720, −4.83680326592193625958493174122, −4.50626503408277851126443800188, −3.01020672966189290262288999198, −1.20612962692361872114232835803, 0.22244817257548982281071561857, 2.30065318040017540018494940035, 3.92414580466833645009298813934, 4.76930104931159190056429381214, 5.16517988128544297545875357749, 6.90007494644420562958420766370, 7.15457813623966246130386270693, 8.521069241150594600331434091957, 9.008969116654277911285328586504, 10.36325742522512368057112797543

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