↑

Properties

Label 2-804-67.40-c1-0-2
Degree $2$
Conductor $804$
Sign $0.347 - 0.937i$
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  + (−0.415 + 0.909i)3-s + (1.80 + 0.528i)5-s + (2.39 + 2.75i)7-s + (−0.654 − 0.755i)9-s + (2.98 + 0.876i)11-s + (0.127 + 0.0817i)13-s + (−1.22 + 1.41i)15-s + (−0.323 − 2.25i)17-s + (−1.17 + 1.35i)19-s + (−3.50 + 1.02i)21-s + (0.274 − 0.600i)23-s + (−1.24 − 0.797i)25-s + (0.959 − 0.281i)27-s + 2.20·29-s + (1.43 − 0.921i)31-s + ⋯
L(s)  = 1  + (−0.239 + 0.525i)3-s + (0.805 + 0.236i)5-s + (0.903 + 1.04i)7-s + (−0.218 − 0.251i)9-s + (0.900 + 0.264i)11-s + (0.0352 + 0.0226i)13-s + (−0.317 + 0.366i)15-s + (−0.0784 − 0.545i)17-s + (−0.268 + 0.310i)19-s + (−0.764 + 0.224i)21-s + (0.0572 − 0.125i)23-s + (−0.248 − 0.159i)25-s + (0.184 − 0.0542i)27-s + 0.409·29-s + (0.257 − 0.165i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.347 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.347 - 0.937i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.347 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50786 + 1.04976i\)
\(L(\frac12)\) \(\approx\) \(1.50786 + 1.04976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-5.35 - 6.19i)T \)
good5 \( 1 + (-1.80 - 0.528i)T + (4.20 + 2.70i)T^{2} \)
7 \( 1 + (-2.39 - 2.75i)T + (-0.996 + 6.92i)T^{2} \)
11 \( 1 + (-2.98 - 0.876i)T + (9.25 + 5.94i)T^{2} \)
13 \( 1 + (-0.127 - 0.0817i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (0.323 + 2.25i)T + (-16.3 + 4.78i)T^{2} \)
19 \( 1 + (1.17 - 1.35i)T + (-2.70 - 18.8i)T^{2} \)
23 \( 1 + (-0.274 + 0.600i)T + (-15.0 - 17.3i)T^{2} \)
29 \( 1 - 2.20T + 29T^{2} \)
31 \( 1 + (-1.43 + 0.921i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + 3.35T + 37T^{2} \)
41 \( 1 + (-1.30 - 9.05i)T + (-39.3 + 11.5i)T^{2} \)
43 \( 1 + (0.0505 + 0.351i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (0.965 - 2.11i)T + (-30.7 - 35.5i)T^{2} \)
53 \( 1 + (-0.154 + 1.07i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-9.65 + 6.20i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (11.7 - 3.44i)T + (51.3 - 32.9i)T^{2} \)
71 \( 1 + (0.518 - 3.60i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (0.182 - 0.0535i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (7.44 + 4.78i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-6.58 - 1.93i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.09 + 2.39i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + 6.01T + 97T^{2} \)
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

−10.30783069787370596167898904688, −9.592480351337666785802021310786, −8.882842991835581784912567053563, −8.059051552324843289826584102052, −6.72772396795278408728576597946, −5.95162581239637787647417878798, −5.13091902763427949311177172655, −4.22600878422129855494367307511, −2.75935037718615124056179764703, −1.66251754437333179785232565063, 1.07221752072737677827771566793, 2.01041912958201983698722073239, 3.72973301688934126622265897946, 4.75010144540147526436997680470, 5.75653148888602596253713862968, 6.63174948778765936892376118953, 7.43905958794383868323115083787, 8.387754219417115435302039548929, 9.177131826184994455693724038544, 10.25350994920248730515679357919

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