↑

Properties

Label 2-804-201.5-c1-0-8
Degree $2$
Conductor $804$
Sign $0.880 + 0.474i$
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  + (−1.56 + 0.749i)3-s + (−3.06 + 0.900i)5-s + (0.591 + 0.512i)7-s + (1.87 − 2.34i)9-s + (−4.86 + 1.42i)11-s + (−2.95 − 4.60i)13-s + (4.11 − 3.70i)15-s + (4.50 + 0.647i)17-s + (1.29 + 1.49i)19-s + (−1.30 − 0.356i)21-s + (2.56 − 1.16i)23-s + (4.38 − 2.81i)25-s + (−1.17 + 5.06i)27-s + 4.91i·29-s + (3.39 − 5.28i)31-s + ⋯
L(s)  = 1  + (−0.901 + 0.432i)3-s + (−1.37 + 0.402i)5-s + (0.223 + 0.193i)7-s + (0.625 − 0.780i)9-s + (−1.46 + 0.430i)11-s + (−0.820 − 1.27i)13-s + (1.06 − 0.956i)15-s + (1.09 + 0.157i)17-s + (0.297 + 0.342i)19-s + (−0.285 − 0.0778i)21-s + (0.533 − 0.243i)23-s + (0.876 − 0.563i)25-s + (−0.225 + 0.974i)27-s + 0.913i·29-s + (0.610 − 0.949i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.880 + 0.474i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.880 + 0.474i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.574237 - 0.144930i\)
\(L(\frac12)\) \(\approx\) \(0.574237 - 0.144930i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.56 - 0.749i)T \)
67 \( 1 + (-1.21 - 8.09i)T \)
good5 \( 1 + (3.06 - 0.900i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-0.591 - 0.512i)T + (0.996 + 6.92i)T^{2} \)
11 \( 1 + (4.86 - 1.42i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (2.95 + 4.60i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-4.50 - 0.647i)T + (16.3 + 4.78i)T^{2} \)
19 \( 1 + (-1.29 - 1.49i)T + (-2.70 + 18.8i)T^{2} \)
23 \( 1 + (-2.56 + 1.16i)T + (15.0 - 17.3i)T^{2} \)
29 \( 1 - 4.91iT - 29T^{2} \)
31 \( 1 + (-3.39 + 5.28i)T + (-12.8 - 28.1i)T^{2} \)
37 \( 1 - 3.77T + 37T^{2} \)
41 \( 1 + (0.190 - 1.32i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (-8.46 - 1.21i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + (1.21 - 0.556i)T + (30.7 - 35.5i)T^{2} \)
53 \( 1 + (1.69 + 11.7i)T + (-50.8 + 14.9i)T^{2} \)
59 \( 1 + (-4.76 + 7.42i)T + (-24.5 - 53.6i)T^{2} \)
61 \( 1 + (-3.83 + 13.0i)T + (-51.3 - 32.9i)T^{2} \)
71 \( 1 + (-4.49 + 0.645i)T + (68.1 - 20.0i)T^{2} \)
73 \( 1 + (-2.01 - 0.591i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (-2.99 - 4.66i)T + (-32.8 + 71.8i)T^{2} \)
83 \( 1 + (-1.16 - 3.95i)T + (-69.8 + 44.8i)T^{2} \)
89 \( 1 + (14.3 + 6.55i)T + (58.2 + 67.2i)T^{2} \)
97 \( 1 - 3.09iT - 97T^{2} \)
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

−10.27997544133059143590607178126, −9.732428852531609731031825619823, −8.044929110115909568986110485730, −7.80110315984372550818483362644, −6.82669023380286254273951088959, −5.43430195200784291666183696761, −5.01939608582199148765691552673, −3.79640969968001460066892024607, −2.81926519152426150867595861578, −0.46506345283286308064197906178, 0.901188887792138594813731284161, 2.71299470564340332392753830367, 4.24076039731921592037900856738, 4.90036590339178336616523486961, 5.81584640079729909043999307401, 7.19998537967047644499351862642, 7.56033367982658192111856214874, 8.335660296426727876241966217193, 9.563594177117801673255370264185, 10.60587692324298211753246922132

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