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Properties

Label 2-804-67.29-c1-0-8
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·5-s + (1.5 + 2.59i)7-s + 9-s + (−1.5 − 2.59i)11-s + (0.5 − 0.866i)13-s + 2·15-s + (−2.5 + 4.33i)17-s + (2.5 − 4.33i)19-s + (−1.5 − 2.59i)21-s + (−4.5 + 7.79i)23-s − 25-s − 27-s + (−4.5 − 7.79i)29-s + (−4.5 − 7.79i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + (0.566 + 0.981i)7-s + 0.333·9-s + (−0.452 − 0.783i)11-s + (0.138 − 0.240i)13-s + 0.516·15-s + (−0.606 + 1.05i)17-s + (0.573 − 0.993i)19-s + (−0.327 − 0.566i)21-s + (−0.938 + 1.62i)23-s − 0.200·25-s − 0.192·27-s + (−0.835 − 1.44i)29-s + (−0.808 − 1.39i)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: $\chi_{804} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.978 + 0.205i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.5 + 7.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (3.5 + 6.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + (7.5 - 12.9i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (3.5 + 6.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.5 - 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.5 + 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (1.5 - 2.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.888194991892945662562729417752, −8.934938935653608923459731528172, −8.021193950610539461483104151201, −7.53113177092967861958031146131, −6.04773228287467491102932943777, −5.59361192158764152527873176146, −4.43201601741551051558351111123, −3.42344995342098223971480031309, −1.94461015212575229558468429613, 0, 1.69207891350361448967366683684, 3.47497722809039265769414895347, 4.46971468745131459722014134272, 5.06341958889950433065933626548, 6.46174331752831049617738794318, 7.38647679068934898745551270851, 7.77331280526901377420835324115, 8.940984862541784357668468073355, 10.03685636903382355903094075904

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