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Properties

Label 2-804-201.200-c1-0-14
Degree $2$
Conductor $804$
Sign $0.780 + 0.624i$
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  + (1.49 − 0.877i)3-s − 1.18·5-s − 0.405i·7-s + (1.46 − 2.62i)9-s + 5.79·11-s + 4.82i·13-s + (−1.77 + 1.04i)15-s − 0.491i·17-s + 3.95·19-s + (−0.355 − 0.605i)21-s − 3.05i·23-s − 3.59·25-s + (−0.116 − 5.19i)27-s − 8.30i·29-s − 5.44i·31-s + ⋯
L(s)  = 1  + (0.862 − 0.506i)3-s − 0.531·5-s − 0.153i·7-s + (0.487 − 0.873i)9-s + 1.74·11-s + 1.33i·13-s + (−0.457 + 0.268i)15-s − 0.119i·17-s + 0.907·19-s + (−0.0775 − 0.132i)21-s − 0.637i·23-s − 0.718·25-s + (−0.0223 − 0.999i)27-s − 1.54i·29-s − 0.977i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.780 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.780 + 0.624i$
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: \((2,\ 804,\ (\ :1/2),\ 0.780 + 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97527 - 0.692822i\)
\(L(\frac12)\) \(\approx\) \(1.97527 - 0.692822i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.49 + 0.877i)T \)
67 \( 1 + (-2.92 - 7.64i)T \)
good5 \( 1 + 1.18T + 5T^{2} \)
7 \( 1 + 0.405iT - 7T^{2} \)
11 \( 1 - 5.79T + 11T^{2} \)
13 \( 1 - 4.82iT - 13T^{2} \)
17 \( 1 + 0.491iT - 17T^{2} \)
19 \( 1 - 3.95T + 19T^{2} \)
23 \( 1 + 3.05iT - 23T^{2} \)
29 \( 1 + 8.30iT - 29T^{2} \)
31 \( 1 + 5.44iT - 31T^{2} \)
37 \( 1 - 0.956T + 37T^{2} \)
41 \( 1 - 6.72T + 41T^{2} \)
43 \( 1 - 7.64iT - 43T^{2} \)
47 \( 1 + 4.74iT - 47T^{2} \)
53 \( 1 + 10.1T + 53T^{2} \)
59 \( 1 + 10.8iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
71 \( 1 - 0.718iT - 71T^{2} \)
73 \( 1 + 9.46T + 73T^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 - 15.8iT - 83T^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−9.660974880292854171356105051519, −9.433507060449215727822345125258, −8.442461297701619686389687627831, −7.61180038952657172658713442107, −6.81766096608404684736571543101, −6.10297778702358750327867305243, −4.20759033736762029755534515019, −3.92873004065201489599942171401, −2.44868706187736837286852260353, −1.18023370790710970867720420699, 1.45907954342793750317946822089, 3.14685444802025765937266797670, 3.69685437067956046589590034516, 4.80392517647477458254953904431, 5.88290687337640734597469140869, 7.19039852464496249520031941801, 7.81182421924803495513539134134, 8.832634653562840316654209835070, 9.308059996317299764892559730033, 10.26593192705890856805312838314

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