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Properties

Label 2-804-201.38-c1-0-17
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: $\chi_{804} (641, \cdot )$
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

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

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