↑

Properties

Label 2-804-201.83-c0-0-0
Degree $2$
Conductor $804$
Sign $0.375 - 0.926i$
Analytic cond. $0.401248$
Root an. cond. $0.633441$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.415 + 0.909i)3-s + (0.0930 + 0.268i)7-s + (−0.654 + 0.755i)9-s + (0.581 + 0.299i)13-s + (−0.271 + 0.785i)19-s + (−0.205 + 0.196i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.419 + 0.216i)31-s + (−0.841 − 1.45i)37-s + (−0.0311 + 0.653i)39-s + (−0.0135 + 0.0941i)43-s + (0.722 − 0.568i)49-s + (−0.827 + 0.0789i)57-s + (−0.469 − 1.93i)61-s + ⋯
L(s)  = 1  + (0.415 + 0.909i)3-s + (0.0930 + 0.268i)7-s + (−0.654 + 0.755i)9-s + (0.581 + 0.299i)13-s + (−0.271 + 0.785i)19-s + (−0.205 + 0.196i)21-s + (0.841 − 0.540i)25-s + (−0.959 − 0.281i)27-s + (−0.419 + 0.216i)31-s + (−0.841 − 1.45i)37-s + (−0.0311 + 0.653i)39-s + (−0.0135 + 0.0941i)43-s + (0.722 − 0.568i)49-s + (−0.827 + 0.0789i)57-s + (−0.469 − 1.93i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.375 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.375 - 0.926i$
Analytic conductor: \(0.401248\)
Root analytic conductor: \(0.633441\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :0),\ 0.375 - 0.926i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.099519802\)
\(L(\frac12)\) \(\approx\) \(1.099519802\)
\(L(1)\) 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 \( 1 + (-0.415 - 0.909i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
good5 \( 1 + (-0.841 + 0.540i)T^{2} \)
7 \( 1 + (-0.0930 - 0.268i)T + (-0.786 + 0.618i)T^{2} \)
11 \( 1 + (0.888 + 0.458i)T^{2} \)
13 \( 1 + (-0.581 - 0.299i)T + (0.580 + 0.814i)T^{2} \)
17 \( 1 + (-0.723 + 0.690i)T^{2} \)
19 \( 1 + (0.271 - 0.785i)T + (-0.786 - 0.618i)T^{2} \)
23 \( 1 + (0.327 + 0.945i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.419 - 0.216i)T + (0.580 - 0.814i)T^{2} \)
37 \( 1 + (0.841 + 1.45i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.235 - 0.971i)T^{2} \)
43 \( 1 + (0.0135 - 0.0941i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (-0.981 - 0.189i)T^{2} \)
53 \( 1 + (0.959 - 0.281i)T^{2} \)
59 \( 1 + (-0.415 - 0.909i)T^{2} \)
61 \( 1 + (0.469 + 1.93i)T + (-0.888 + 0.458i)T^{2} \)
71 \( 1 + (-0.723 - 0.690i)T^{2} \)
73 \( 1 + (-0.341 - 1.40i)T + (-0.888 + 0.458i)T^{2} \)
79 \( 1 + (0.0845 + 1.77i)T + (-0.995 + 0.0950i)T^{2} \)
83 \( 1 + (-0.0475 - 0.998i)T^{2} \)
89 \( 1 + (0.654 + 0.755i)T^{2} \)
97 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.58511568695842468281060859720, −9.786083780400397514593249486352, −8.858024482374829284518858143755, −8.400160822301562261293309017334, −7.29314323794317086238352700185, −6.09898794166122944992746323428, −5.21799120899252185514025964947, −4.19368305421046691776372819097, −3.32947025857748686104085514023, −2.05026969743191707331917852242, 1.25976444810268289133807292541, 2.64720953056814666169172162936, 3.66563119622807271393291831486, 5.00066556384378405651736704725, 6.13129602740395397393690031606, 6.92951711218432354750565652113, 7.69547051543497394623257634230, 8.605183402139510990682026590787, 9.189690328088414094026382107449, 10.39642409709426504521439139552

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