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Properties

Label 2-804-201.38-c1-0-9
Degree $2$
Conductor $804$
Sign $0.952 - 0.304i$
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  + (−0.355 + 1.69i)3-s − 3.60·5-s + (3.20 − 1.84i)7-s + (−2.74 − 1.20i)9-s + (−1.48 − 2.56i)11-s + (1.95 + 1.12i)13-s + (1.27 − 6.10i)15-s + (4.00 + 2.30i)17-s + (2.85 − 4.94i)19-s + (1.99 + 6.08i)21-s + (7.12 + 4.11i)23-s + 7.98·25-s + (3.01 − 4.23i)27-s + (−4.16 + 2.40i)29-s + (−6.73 + 3.89i)31-s + ⋯
L(s)  = 1  + (−0.205 + 0.978i)3-s − 1.61·5-s + (1.20 − 0.698i)7-s + (−0.915 − 0.401i)9-s + (−0.446 − 0.773i)11-s + (0.541 + 0.312i)13-s + (0.330 − 1.57i)15-s + (0.970 + 0.560i)17-s + (0.655 − 1.13i)19-s + (0.435 + 1.32i)21-s + (1.48 + 0.857i)23-s + 1.59·25-s + (0.580 − 0.814i)27-s + (−0.774 + 0.447i)29-s + (−1.21 + 0.698i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.952 - 0.304i$
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: \((2,\ 804,\ (\ :1/2),\ 0.952 - 0.304i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16860 + 0.182113i\)
\(L(\frac12)\) \(\approx\) \(1.16860 + 0.182113i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.355 - 1.69i)T \)
67 \( 1 + (-2.48 - 7.79i)T \)
good5 \( 1 + 3.60T + 5T^{2} \)
7 \( 1 + (-3.20 + 1.84i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.48 + 2.56i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.95 - 1.12i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-4.00 - 2.30i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.85 + 4.94i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.12 - 4.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.16 - 2.40i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.73 - 3.89i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.09 + 7.08i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.243 - 0.422i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + (-7.37 + 4.25i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 + 11.3iT - 59T^{2} \)
61 \( 1 + (3.97 + 2.29i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (-12.3 + 7.13i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.629 + 1.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.80 + 2.19i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.80 - 1.62i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.12iT - 89T^{2} \)
97 \( 1 + (-8.97 - 5.18i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.77947580840081464698216169204, −9.341776986481379901013675332541, −8.571658073885802482582471550382, −7.78777711438123094054919945110, −7.14874930732529671084266626860, −5.51505750398828480479949350754, −4.83713130756105571556495175715, −3.84024378317085867749480750454, −3.28516009971322061200084412454, −0.863692356803301718976705095567, 1.01149361467465870112705245453, 2.46602235577132844710964233707, 3.74309226494166248699617832744, 5.00247588068077361723892236392, 5.66717155819874319830230330873, 7.16890850201742026131632407316, 7.67398131376950288656320730230, 8.179994632632722311976027562502, 9.034360977442598407117834835053, 10.56723293209175695157591160879

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