↑

Properties

Label 2-804-12.11-c1-0-89
Degree $2$
Conductor $804$
Sign $-0.128 + 0.991i$
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  + (−1.28 − 0.587i)2-s + (1.29 − 1.15i)3-s + (1.31 + 1.51i)4-s + 1.10i·5-s + (−2.34 + 0.721i)6-s − 0.262i·7-s + (−0.798 − 2.71i)8-s + (0.347 − 2.97i)9-s + (0.651 − 1.42i)10-s − 2.52·11-s + (3.43 + 0.446i)12-s + 2.54·13-s + (−0.154 + 0.337i)14-s + (1.27 + 1.43i)15-s + (−0.566 + 3.95i)16-s − 1.62i·17-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.746 − 0.664i)3-s + (0.655 + 0.755i)4-s + 0.496i·5-s + (−0.955 + 0.294i)6-s − 0.0992i·7-s + (−0.282 − 0.959i)8-s + (0.115 − 0.993i)9-s + (0.206 − 0.451i)10-s − 0.762·11-s + (0.991 + 0.128i)12-s + 0.705·13-s + (−0.0412 + 0.0902i)14-s + (0.329 + 0.370i)15-s + (−0.141 + 0.989i)16-s − 0.394i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802432 - 0.913393i\)
\(L(\frac12)\) \(\approx\) \(0.802432 - 0.913393i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 0.587i)T \)
3 \( 1 + (-1.29 + 1.15i)T \)
67 \( 1 + iT \)
good5 \( 1 - 1.10iT - 5T^{2} \)
7 \( 1 + 0.262iT - 7T^{2} \)
11 \( 1 + 2.52T + 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 + 1.62iT - 17T^{2} \)
19 \( 1 + 7.27iT - 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 + 2.37iT - 29T^{2} \)
31 \( 1 + 4.13iT - 31T^{2} \)
37 \( 1 - 9.94T + 37T^{2} \)
41 \( 1 + 0.158iT - 41T^{2} \)
43 \( 1 - 0.426iT - 43T^{2} \)
47 \( 1 - 6.49T + 47T^{2} \)
53 \( 1 + 2.11iT - 53T^{2} \)
59 \( 1 - 1.36T + 59T^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
71 \( 1 - 4.23T + 71T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 - 11.8iT - 79T^{2} \)
83 \( 1 - 0.252T + 83T^{2} \)
89 \( 1 + 7.08iT - 89T^{2} \)
97 \( 1 + 9.60T + 97T^{2} \)
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

−9.889981333705532170016135207535, −9.127648362140471222233922923101, −8.383724577154909334407518453858, −7.55634618888589772791426044906, −6.95373474795190637823068475470, −6.01759799803011603700224821563, −4.21280528627413465856880140888, −2.95213225833373736110364579003, −2.37632171948198305204455211252, −0.78182222797194692793724198778, 1.51281894172648581303269803014, 2.81719781548063213495721138575, 4.12661791803551032378924580890, 5.31012993121356298651809691557, 6.08437257709834236225329171866, 7.45937093143323691463574391396, 8.164803784421886441747496107019, 8.713385187935151925336139374307, 9.496934325315633717760774775073, 10.42451520969099536835470217842

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