↑

Properties

Label 2-804-67.37-c1-0-0
Degree $2$
Conductor $804$
Sign $-0.978 - 0.205i$
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  − 3-s + 2·5-s + (−2.5 + 4.33i)7-s + 9-s + (−2.5 + 4.33i)11-s + (−1.5 − 2.59i)13-s − 2·15-s + (−1.5 − 2.59i)17-s + (−3.5 − 6.06i)19-s + (2.5 − 4.33i)21-s + (−1.5 − 2.59i)23-s − 25-s − 27-s + (0.5 − 0.866i)29-s + (−2.5 + 4.33i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + (−0.944 + 1.63i)7-s + 0.333·9-s + (−0.753 + 1.30i)11-s + (−0.416 − 0.720i)13-s − 0.516·15-s + (−0.363 − 0.630i)17-s + (−0.802 − 1.39i)19-s + (0.545 − 0.944i)21-s + (−0.312 − 0.541i)23-s − 0.200·25-s − 0.192·27-s + (0.0928 − 0.160i)29-s + (−0.449 + 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.978 - 0.205i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.978 - 0.205i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0437024 + 0.420896i\)
\(L(\frac12)\) \(\approx\) \(0.0437024 + 0.420896i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
67 \( 1 + (-8 + 1.73i)T \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (2.5 - 4.33i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.5 + 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.5 - 9.52i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-1.5 + 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.5 - 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.47330179768290229964536643990, −9.716077733487740726662263683089, −9.289455297785225072877052958819, −8.189151490038912072878799810193, −6.88616679477676019542069509815, −6.28770276629955051138768216279, −5.32449758289253266854876728478, −4.77566353187544674384816648908, −2.78425961494884892274306576719, −2.21218331724074080016703951081, 0.20617530664413012890178828043, 1.84417672452927925837383579503, 3.49342363351217763007779990538, 4.28799819643648618928183552791, 5.77039090377263434220253935699, 6.15836565015958488463690392088, 7.14709394731671295485955705971, 8.028178539005413540804408531274, 9.277037451098849523596741938201, 10.07222222908808503930706382522

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