Properties

Label 2-804-201.38-c1-0-8
Degree $2$
Conductor $804$
Sign $0.194 - 0.980i$
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 + 1.41i)3-s + 2·5-s + (−0.949 + 0.548i)7-s + (−1.00 − 2.82i)9-s + (−0.724 − 1.25i)11-s + (4.5 + 2.59i)13-s + (−2 + 2.82i)15-s + (5.17 + 2.98i)17-s + (−1.5 + 2.59i)19-s + (0.174 − 1.89i)21-s + (0.825 + 0.476i)23-s − 25-s + (5.00 + 1.41i)27-s + (4.62 − 2.66i)29-s + (−1.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.577 + 0.816i)3-s + 0.894·5-s + (−0.358 + 0.207i)7-s + (−0.333 − 0.942i)9-s + (−0.218 − 0.378i)11-s + (1.24 + 0.720i)13-s + (−0.516 + 0.730i)15-s + (1.25 + 0.724i)17-s + (−0.344 + 0.596i)19-s + (0.0380 − 0.412i)21-s + (0.172 + 0.0994i)23-s − 0.200·25-s + (0.962 + 0.272i)27-s + (0.858 − 0.495i)29-s + (−0.269 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.194 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.194 - 0.980i$
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: \((2,\ 804,\ (\ :1/2),\ 0.194 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10130 + 0.904747i\)
\(L(\frac12)\) \(\approx\) \(1.10130 + 0.904747i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.41i)T \)
67 \( 1 + (-8 + 1.73i)T \)
good5 \( 1 - 2T + 5T^{2} \)
7 \( 1 + (0.949 - 0.548i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.724 + 1.25i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-5.17 - 2.98i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 - 2.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.825 - 0.476i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.62 + 2.66i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.94 - 8.57i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.72 - 2.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 1.27iT - 43T^{2} \)
47 \( 1 + (-0.275 + 0.158i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 0.635iT - 59T^{2} \)
61 \( 1 + (-9.39 - 5.42i)T + (30.5 + 52.8i)T^{2} \)
71 \( 1 + (11.1 - 6.45i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.94 + 5.10i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.398 + 0.230i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-10.0 - 5.81i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 4.73iT - 89T^{2} \)
97 \( 1 + (-11.8 - 6.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.26924502168440954602458258485, −9.830711304106414275012054924919, −8.906772354130424201954381808005, −8.133890020504848344636050079788, −6.49024574580801127004471546754, −6.05893219597468323138153944893, −5.28733761923950854152890524689, −4.04681645775054943604972298793, −3.14109714642248569359452829589, −1.43236094724480744087534942019, 0.853877751137873816432974287272, 2.15733390315556777864451961461, 3.40355788144005033423712426126, 5.05190018588549544553177579666, 5.73224068729256377761131635216, 6.52153045756498112884133662249, 7.36299024592476012280705136459, 8.262462244150153419153292713065, 9.283499877331238913468501405231, 10.25276823656645242545580683224

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