Properties

Label 2-804-201.164-c1-0-4
Degree $2$
Conductor $804$
Sign $-0.456 - 0.889i$
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  + (−0.869 + 1.49i)3-s + 2.66·5-s + (0.767 + 0.442i)7-s + (−1.48 − 2.60i)9-s + (−2.39 + 4.15i)11-s + (−5.90 + 3.40i)13-s + (−2.32 + 3.99i)15-s + (5.55 − 3.20i)17-s + (2.50 + 4.33i)19-s + (−1.33 + 0.764i)21-s + (−1.46 + 0.846i)23-s + 2.12·25-s + (5.19 + 0.0333i)27-s + (5.11 + 2.95i)29-s + (−5.28 − 3.05i)31-s + ⋯
L(s)  = 1  + (−0.501 + 0.864i)3-s + 1.19·5-s + (0.289 + 0.167i)7-s + (−0.496 − 0.868i)9-s + (−0.722 + 1.25i)11-s + (−1.63 + 0.944i)13-s + (−0.599 + 1.03i)15-s + (1.34 − 0.778i)17-s + (0.573 + 0.994i)19-s + (−0.290 + 0.166i)21-s + (−0.305 + 0.176i)23-s + 0.425·25-s + (0.999 + 0.00641i)27-s + (0.949 + 0.548i)29-s + (−0.949 − 0.548i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.456 - 0.889i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684834 + 1.12064i\)
\(L(\frac12)\) \(\approx\) \(0.684834 + 1.12064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.869 - 1.49i)T \)
67 \( 1 + (-2.91 + 7.64i)T \)
good5 \( 1 - 2.66T + 5T^{2} \)
7 \( 1 + (-0.767 - 0.442i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.39 - 4.15i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.90 - 3.40i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-5.55 + 3.20i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.50 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.46 - 0.846i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.11 - 2.95i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.28 + 3.05i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.639 - 1.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.45 - 2.51i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 + (-3.58 - 2.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.25T + 53T^{2} \)
59 \( 1 + 0.587iT - 59T^{2} \)
61 \( 1 + (5.85 - 3.38i)T + (30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.38 - 0.797i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.71 - 6.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.0 - 5.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.12 - 4.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.09iT - 89T^{2} \)
97 \( 1 + (-7.83 + 4.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.08334393947750047575253046916, −9.797050195293373143609574236929, −9.427775919752273183441760069837, −7.907424881133614167553844143853, −7.03750595609253165925267918572, −5.88378022489003738360059439405, −5.13124632818594058763738660694, −4.57571420632529107027157359807, −2.97407507642764100186030293292, −1.81052436658091292735434450138, 0.67814778560607236356638145679, 2.13722143070344701372976821278, 3.08161741941892508536348930791, 5.18017810061075546338207504830, 5.45364249344152728421327131679, 6.35664468804858083755303576697, 7.50043611644381050368925251010, 8.018932123557715086971377949757, 9.147942319858852757591781467442, 10.37194288974927884093027534279

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