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Properties

Label 2-804-201.38-c1-0-5
Degree $2$
Conductor $804$
Sign $0.826 + 0.563i$
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  + (−1.34 + 1.09i)3-s − 3.88·5-s + (−3.35 + 1.93i)7-s + (0.618 − 2.93i)9-s + (0.627 + 1.08i)11-s + (0.212 + 0.122i)13-s + (5.22 − 4.23i)15-s + (−2.38 − 1.37i)17-s + (−2.25 + 3.90i)19-s + (2.40 − 6.27i)21-s + (−2.10 − 1.21i)23-s + 10.0·25-s + (2.37 + 4.62i)27-s + (3.32 − 1.91i)29-s + (4.36 − 2.51i)31-s + ⋯
L(s)  = 1  + (−0.776 + 0.629i)3-s − 1.73·5-s + (−1.26 + 0.733i)7-s + (0.206 − 0.978i)9-s + (0.189 + 0.327i)11-s + (0.0589 + 0.0340i)13-s + (1.34 − 1.09i)15-s + (−0.577 − 0.333i)17-s + (−0.517 + 0.895i)19-s + (0.524 − 1.36i)21-s + (−0.439 − 0.253i)23-s + 2.01·25-s + (0.456 + 0.889i)27-s + (0.617 − 0.356i)29-s + (0.783 − 0.452i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.826 + 0.563i$
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: \((2,\ 804,\ (\ :1/2),\ 0.826 + 0.563i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.323080 - 0.0996092i\)
\(L(\frac12)\) \(\approx\) \(0.323080 - 0.0996092i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.34 - 1.09i)T \)
67 \( 1 + (-0.688 - 8.15i)T \)
good5 \( 1 + 3.88T + 5T^{2} \)
7 \( 1 + (3.35 - 1.93i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.627 - 1.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.212 - 0.122i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.38 + 1.37i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.25 - 3.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.10 + 1.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.32 + 1.91i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.36 + 2.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.49 + 7.78i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.59 - 6.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 4.98iT - 43T^{2} \)
47 \( 1 + (-4.99 + 2.88i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 13.5iT - 59T^{2} \)
61 \( 1 + (6.26 + 3.61i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-4.57 + 2.64i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.33 - 2.31i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-10.3 + 5.96i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.48 + 2.59i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 14.5iT - 89T^{2} \)
97 \( 1 + (7.87 + 4.54i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.23603104557986155731029917589, −9.388302139692467144497619822872, −8.587583593793342200273397192682, −7.55395318716612645707917789963, −6.56282186983780982423560873455, −5.88854287540643030145739492015, −4.48772065202881880368314531982, −3.96136494368469585756952907886, −2.88506830641948223380438351612, −0.28966698811300635692477078316, 0.78145196923606375937833113019, 2.95338975895548812364127302284, 4.00410427613252889995008044339, 4.79838750795061766753779604516, 6.43666463146949706405839028962, 6.69537335124957156977808950899, 7.70525158874404127922733451772, 8.343184805830418375182470721214, 9.550777201068659714469823134193, 10.77789186201782372118800617208

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