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Properties

Label 2-804-201.200-c1-0-13
Degree $2$
Conductor $804$
Sign $0.992 + 0.124i$
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  + (−1.58 + 0.707i)3-s + 3.16·5-s − 1.41i·7-s + (2.00 − 2.23i)9-s + 3.16·11-s + 4.24i·13-s + (−5.00 + 2.23i)15-s − 6.70i·17-s − 3·19-s + (1.00 + 2.23i)21-s − 6.70i·23-s + 5.00·25-s + (−1.58 + 4.94i)27-s + 2.23i·29-s − 4.24i·31-s + ⋯
L(s)  = 1  + (−0.912 + 0.408i)3-s + 1.41·5-s − 0.534i·7-s + (0.666 − 0.745i)9-s + 0.953·11-s + 1.17i·13-s + (−1.29 + 0.577i)15-s − 1.62i·17-s − 0.688·19-s + (0.218 + 0.487i)21-s − 1.39i·23-s + 1.00·25-s + (−0.304 + 0.952i)27-s + 0.415i·29-s − 0.762i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.992 + 0.124i$
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: \((2,\ 804,\ (\ :1/2),\ 0.992 + 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54881 - 0.0964237i\)
\(L(\frac12)\) \(\approx\) \(1.54881 - 0.0964237i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 - 0.707i)T \)
67 \( 1 + (7 + 4.24i)T \)
good5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 + 1.41iT - 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 - 4.24iT - 13T^{2} \)
17 \( 1 + 6.70iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 + 6.70iT - 23T^{2} \)
29 \( 1 - 2.23iT - 29T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 12.6T + 41T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 - 2.23iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 2.23iT - 59T^{2} \)
61 \( 1 + 2.82iT - 61T^{2} \)
71 \( 1 - 13.4iT - 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 + 14.1iT - 79T^{2} \)
83 \( 1 + 4.47iT - 83T^{2} \)
89 \( 1 + 2.23iT - 89T^{2} \)
97 \( 1 - 8.48iT - 97T^{2} \)
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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

−10.19414330895103725277636173339, −9.390089349175925021436380624132, −9.105108704153527324203743589833, −7.34822885380022822483018009780, −6.43960915658412676994465965770, −6.07826747486617672411190416660, −4.77892523546812327037685112636, −4.20014960372126109493567815418, −2.45590510625300637903520791201, −1.05232544094010160938320677403, 1.31851204504182609025724914495, 2.29009983163835698543836546844, 3.95467863585309783018298840209, 5.37945793692083760899248127196, 5.88143900753117517513427491639, 6.43563795276384791839368163809, 7.60107474010325987211904997078, 8.669382773459439844681958704413, 9.567324188810479551833918946478, 10.37725068281455697535919945455

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