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Properties

Label 2-804-201.200-c1-0-7
Degree $2$
Conductor $804$
Sign $0.999 - 0.0122i$
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  + (−0.760 + 1.55i)3-s − 3.48·5-s − 1.71i·7-s + (−1.84 − 2.36i)9-s + 0.624·11-s + 2.24i·13-s + (2.65 − 5.42i)15-s − 3.10i·17-s + 5.86·19-s + (2.67 + 1.30i)21-s + 1.75i·23-s + 7.17·25-s + (5.08 − 1.06i)27-s − 0.0875i·29-s + 0.222i·31-s + ⋯
L(s)  = 1  + (−0.439 + 0.898i)3-s − 1.56·5-s − 0.649i·7-s + (−0.614 − 0.789i)9-s + 0.188·11-s + 0.623i·13-s + (0.685 − 1.40i)15-s − 0.753i·17-s + 1.34·19-s + (0.583 + 0.285i)21-s + 0.365i·23-s + 1.43·25-s + (0.978 − 0.205i)27-s − 0.0162i·29-s + 0.0399i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.999 - 0.0122i$
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: \((2,\ 804,\ (\ :1/2),\ 0.999 - 0.0122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864604 + 0.00530706i\)
\(L(\frac12)\) \(\approx\) \(0.864604 + 0.00530706i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.760 - 1.55i)T \)
67 \( 1 + (3.68 + 7.30i)T \)
good5 \( 1 + 3.48T + 5T^{2} \)
7 \( 1 + 1.71iT - 7T^{2} \)
11 \( 1 - 0.624T + 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 - 5.86T + 19T^{2} \)
23 \( 1 - 1.75iT - 23T^{2} \)
29 \( 1 + 0.0875iT - 29T^{2} \)
31 \( 1 - 0.222iT - 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 - 5.27T + 41T^{2} \)
43 \( 1 + 7.30iT - 43T^{2} \)
47 \( 1 + 1.62iT - 47T^{2} \)
53 \( 1 - 7.57T + 53T^{2} \)
59 \( 1 - 4.77iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
71 \( 1 + 5.72iT - 71T^{2} \)
73 \( 1 - 5.99T + 73T^{2} \)
79 \( 1 + 8.28iT - 79T^{2} \)
83 \( 1 - 8.79iT - 83T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 + 18.0iT - 97T^{2} \)
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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

−10.36453009185127252445629573305, −9.444082899028409030276666059752, −8.684680754359962356449100178352, −7.52252529230479027913322284849, −7.05215775455997603897071120083, −5.67734571209320966738415792897, −4.60259341784749157460953774688, −3.97697212405801637783462244544, −3.14079290746133291551133621672, −0.65885202884377595174541444497, 0.923781109163994766785695514442, 2.64772593892798157667468937606, 3.76404820277797024274360886107, 4.99107110920153253273717817786, 5.91981927713398712648710566828, 6.91553020639517884019122176541, 7.83851698590103445010775711855, 8.153433659239636470852382556478, 9.231690646533581388368321315536, 10.54925438577248622456395573767

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