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Properties

Label 2-804-201.200-c1-0-11
Degree $2$
Conductor $804$
Sign $0.770 - 0.637i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.770 - 0.637i$
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: \((2,\ 804,\ (\ :1/2),\ 0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39520 + 0.862323i\)
\(L(\frac12)\) \(\approx\) \(2.39520 + 0.862323i\)
\(L(\frac{3}{2})\) 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

−10.15375896012206083909900463513, −9.350820016449875475829063859443, −8.757770004293140423631172769354, −8.116614509083773716533802419642, −6.79935187105351517699213095238, −5.99339110244929193166701985794, −5.04948263896568375212179185887, −3.81430644351178491481346939976, −2.40115315340786020459770343925, −2.01608585232388351683009257272, 1.27700966497499060363781127523, 2.49655848217785737193262325200, 3.59474174771469984200069281468, 4.62690076699224570690274397432, 5.78702372908487965111209482676, 6.96066218694083662959442106371, 7.51823705343184607794173756265, 8.521477512458227069506939280516, 9.392902771106922349405676816681, 10.18713123618936924745022124421

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