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Properties

Label 2-804-12.11-c1-0-109
Degree $2$
Conductor $804$
Sign $-0.965 + 0.261i$
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  + (−1.37 + 0.330i)2-s + (0.355 − 1.69i)3-s + (1.78 − 0.907i)4-s − 0.0453i·5-s + (0.0712 + 2.44i)6-s − 2.82i·7-s + (−2.15 + 1.83i)8-s + (−2.74 − 1.20i)9-s + (0.0149 + 0.0623i)10-s + 3.33·11-s + (−0.906 − 3.34i)12-s − 5.31·13-s + (0.931 + 3.88i)14-s + (−0.0768 − 0.0160i)15-s + (2.35 − 3.23i)16-s − 0.0462i·17-s + ⋯
L(s)  = 1  + (−0.972 + 0.233i)2-s + (0.205 − 0.978i)3-s + (0.891 − 0.453i)4-s − 0.0202i·5-s + (0.0291 + 0.999i)6-s − 1.06i·7-s + (−0.760 + 0.649i)8-s + (−0.915 − 0.401i)9-s + (0.00473 + 0.0197i)10-s + 1.00·11-s + (−0.261 − 0.965i)12-s − 1.47·13-s + (0.248 + 1.03i)14-s + (−0.0198 − 0.00415i)15-s + (0.587 − 0.808i)16-s − 0.0112i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.965 + 0.261i$
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: \((2,\ 804,\ (\ :1/2),\ -0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0840798 - 0.631565i\)
\(L(\frac12)\) \(\approx\) \(0.0840798 - 0.631565i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.330i)T \)
3 \( 1 + (-0.355 + 1.69i)T \)
67 \( 1 - iT \)
good5 \( 1 + 0.0453iT - 5T^{2} \)
7 \( 1 + 2.82iT - 7T^{2} \)
11 \( 1 - 3.33T + 11T^{2} \)
13 \( 1 + 5.31T + 13T^{2} \)
17 \( 1 + 0.0462iT - 17T^{2} \)
19 \( 1 + 3.00iT - 19T^{2} \)
23 \( 1 + 8.74T + 23T^{2} \)
29 \( 1 + 3.33iT - 29T^{2} \)
31 \( 1 + 1.84iT - 31T^{2} \)
37 \( 1 - 0.528T + 37T^{2} \)
41 \( 1 - 4.07iT - 41T^{2} \)
43 \( 1 + 0.346iT - 43T^{2} \)
47 \( 1 + 0.457T + 47T^{2} \)
53 \( 1 - 2.89iT - 53T^{2} \)
59 \( 1 + 0.226T + 59T^{2} \)
61 \( 1 + 6.11T + 61T^{2} \)
71 \( 1 + 7.18T + 71T^{2} \)
73 \( 1 + 5.00T + 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 - 0.699iT - 89T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−9.747438838454848376599061526165, −8.986149928396064024317357459416, −7.987946673477227890305881338354, −7.34855146599819802066211940681, −6.75674470454890978261740685329, −5.89786266903509553639997367717, −4.40664889457371179137151050400, −2.86758025683588829198909834798, −1.69533432001794403377944158409, −0.40294677347309877985589452977, 2.00553924839900910493336308619, 2.99132442653024837924772528622, 4.13157108592703585896773820867, 5.39390359725953963835226620700, 6.32587390283505560585832242695, 7.47668904570346901834466645147, 8.476018207994173040280116941591, 9.023811027914385505994920453411, 9.800644141145306826683405525835, 10.29633859167205447596810446351

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