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Properties

Label 2-804-67.25-c1-0-9
Degree $2$
Conductor $804$
Sign $0.737 + 0.675i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.654 − 0.755i)3-s + (2.59 − 1.66i)5-s + (0.180 + 1.25i)7-s + (−0.142 − 0.989i)9-s + (4.04 − 2.60i)11-s + (−2.03 + 4.45i)13-s + (0.438 − 3.05i)15-s + (3.53 + 1.03i)17-s + (0.417 − 2.90i)19-s + (1.06 + 0.687i)21-s + (−1.78 + 2.05i)23-s + (1.87 − 4.09i)25-s + (−0.841 − 0.540i)27-s + 1.04·29-s + (−1.99 − 4.36i)31-s + ⋯
L(s)  = 1  + (0.378 − 0.436i)3-s + (1.15 − 0.745i)5-s + (0.0684 + 0.475i)7-s + (−0.0474 − 0.329i)9-s + (1.22 − 0.784i)11-s + (−0.563 + 1.23i)13-s + (0.113 − 0.787i)15-s + (0.856 + 0.251i)17-s + (0.0957 − 0.665i)19-s + (0.233 + 0.150i)21-s + (−0.371 + 0.428i)23-s + (0.374 − 0.819i)25-s + (−0.161 − 0.104i)27-s + 0.193·29-s + (−0.358 − 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09511 - 0.814396i\)
\(L(\frac12)\) \(\approx\) \(2.09511 - 0.814396i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (5.33 + 6.20i)T \)
good5 \( 1 + (-2.59 + 1.66i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.180 - 1.25i)T + (-6.71 + 1.97i)T^{2} \)
11 \( 1 + (-4.04 + 2.60i)T + (4.56 - 10.0i)T^{2} \)
13 \( 1 + (2.03 - 4.45i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (-3.53 - 1.03i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (-0.417 + 2.90i)T + (-18.2 - 5.35i)T^{2} \)
23 \( 1 + (1.78 - 2.05i)T + (-3.27 - 22.7i)T^{2} \)
29 \( 1 - 1.04T + 29T^{2} \)
31 \( 1 + (1.99 + 4.36i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + 7.59T + 37T^{2} \)
41 \( 1 + (4.78 + 1.40i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-10.2 - 3.01i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + (0.421 - 0.486i)T + (-6.68 - 46.5i)T^{2} \)
53 \( 1 + (13.4 - 3.94i)T + (44.5 - 28.6i)T^{2} \)
59 \( 1 + (1.53 + 3.36i)T + (-38.6 + 44.5i)T^{2} \)
61 \( 1 + (0.360 + 0.231i)T + (25.3 + 55.4i)T^{2} \)
71 \( 1 + (-9.46 + 2.77i)T + (59.7 - 38.3i)T^{2} \)
73 \( 1 + (0.916 + 0.588i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-1.64 + 3.59i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (4.80 - 3.08i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-8.42 - 9.72i)T + (-12.6 + 88.0i)T^{2} \)
97 \( 1 + 5.06T + 97T^{2} \)
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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

−9.765716109606039434892815806563, −9.203593871202014802164852746221, −8.787756688937998484836206390061, −7.62558397042913850071529829949, −6.51231315929032014373762694483, −5.88627198821260063783608443556, −4.88479329351054722135920101540, −3.62615468031599338372955747801, −2.20406535762511076773310425764, −1.31463527747210795651587033274, 1.60645805607820808586524953924, 2.83524858420446076199440721399, 3.81315383512291009192063667306, 5.04968370232723396730534110300, 5.95966013246282580688467945024, 6.93415896526020209228290526475, 7.69678232149885073765786275354, 8.848909629688606645946993585065, 9.835807071548024272600202144455, 10.11781750765384767841681311911

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