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Properties

Label 2-804-201.200-c1-0-12
Degree $2$
Conductor $804$
Sign $-0.679 + 0.733i$
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  + (−1.72 + 0.114i)3-s − 2.40·5-s + 3.48i·7-s + (2.97 − 0.396i)9-s + 1.00·11-s + 1.58i·13-s + (4.15 − 0.276i)15-s − 3.02i·17-s − 3.52·19-s + (−0.399 − 6.01i)21-s + 5.89i·23-s + 0.782·25-s + (−5.09 + 1.02i)27-s − 6.29i·29-s − 10.0i·31-s + ⋯
L(s)  = 1  + (−0.997 + 0.0662i)3-s − 1.07·5-s + 1.31i·7-s + (0.991 − 0.132i)9-s + 0.303·11-s + 0.440i·13-s + (1.07 − 0.0712i)15-s − 0.734i·17-s − 0.809·19-s + (−0.0871 − 1.31i)21-s + 1.22i·23-s + 0.156·25-s + (−0.980 + 0.197i)27-s − 1.16i·29-s − 1.80i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.679 + 0.733i$
Analytic conductor: \(6.41997\)
Root analytic conductor: \(2.53376\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 804,\ (\ :1/2),\ -0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0425021 - 0.0972842i\)
\(L(\frac12)\) \(\approx\) \(0.0425021 - 0.0972842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.114i)T \)
67 \( 1 + (-5.94 + 5.62i)T \)
good5 \( 1 + 2.40T + 5T^{2} \)
7 \( 1 - 3.48iT - 7T^{2} \)
11 \( 1 - 1.00T + 11T^{2} \)
13 \( 1 - 1.58iT - 13T^{2} \)
17 \( 1 + 3.02iT - 17T^{2} \)
19 \( 1 + 3.52T + 19T^{2} \)
23 \( 1 - 5.89iT - 23T^{2} \)
29 \( 1 + 6.29iT - 29T^{2} \)
31 \( 1 + 10.0iT - 31T^{2} \)
37 \( 1 + 6.52T + 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 + 5.62iT - 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 - 2.62iT - 59T^{2} \)
61 \( 1 + 12.1iT - 61T^{2} \)
71 \( 1 - 9.00iT - 71T^{2} \)
73 \( 1 + 0.635T + 73T^{2} \)
79 \( 1 - 9.44iT - 79T^{2} \)
83 \( 1 + 7.80iT - 83T^{2} \)
89 \( 1 + 15.8iT - 89T^{2} \)
97 \( 1 + 8.49iT - 97T^{2} \)
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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

−9.886324299037425240559058759535, −9.211271698474252617064464098601, −8.174190886600611499872681219413, −7.32389340072998625177892424035, −6.33216413605552415835435583277, −5.54742746454894577741704113141, −4.56651377582804502641230014328, −3.64288733351244075753043948866, −2.04396735530331126998957971684, −0.06521495081032579798473533254, 1.30976863130410684087668482149, 3.49524353521968498130344578411, 4.26636481528942887720870684245, 5.07017181301237626008995423033, 6.48366509103023631256323417757, 6.97233240447571824587304792451, 7.88279118981402043285488390138, 8.705017652059221855808600648155, 10.20814675212368360895727781418, 10.62582760881833210491856983971

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