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Properties

Label 2-804-201.164-c1-0-22
Degree $2$
Conductor $804$
Sign $-0.586 - 0.809i$
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  − 1.73i·3-s + (−3 − 1.73i)7-s − 2.99·9-s + (−4.5 + 2.59i)13-s + (4 + 6.92i)19-s + (−2.99 + 5.19i)21-s − 5·25-s + 5.19i·27-s + (1.5 + 0.866i)31-s + (−5 − 8.66i)37-s + (4.5 + 7.79i)39-s − 1.73i·43-s + (2.5 + 4.33i)49-s + (11.9 − 6.92i)57-s + (−13.5 + 7.79i)61-s + ⋯
L(s)  = 1  − 0.999i·3-s + (−1.13 − 0.654i)7-s − 0.999·9-s + (−1.24 + 0.720i)13-s + (0.917 + 1.58i)19-s + (−0.654 + 1.13i)21-s − 25-s + 0.999i·27-s + (0.269 + 0.155i)31-s + (−0.821 − 1.42i)37-s + (0.720 + 1.24i)39-s − 0.264i·43-s + (0.357 + 0.618i)49-s + (1.58 − 0.917i)57-s + (−1.72 + 0.997i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.586 - 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.586 - 0.809i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.586 - 0.809i)\)

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 \( 1 + 1.73iT \)
67 \( 1 + (5.5 - 6.06i)T \)
good5 \( 1 + 5T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.5 - 2.59i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4 - 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (13.5 - 7.79i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (16.5 - 9.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.753262822512613540176371550607, −8.889363340673095284466500425189, −7.60658930531610968604918821978, −7.28615047395844851150059915793, −6.30624219427902568529593998733, −5.50974263582074932377974302382, −4.03518216850404759940132545039, −2.98916828815000323641775569873, −1.69955755213743297015120101622, 0, 2.67597552725336831600453735092, 3.24683755246149025095292081906, 4.64328377949172704156158786751, 5.38562577314389272330452197398, 6.31460259250333074662458331719, 7.38647807939077263542889296527, 8.473594594990375988546197641721, 9.520509593862461426931722994725, 9.669008988730103615969072931109

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