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Properties

Label 2-805-805.114-c0-0-0
Degree $2$
Conductor $805$
Sign $-0.0633 - 0.997i$
Analytic cond. $0.401747$
Root an. cond. $0.633835$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)17-s + 0.999·20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·28-s − 29-s + (0.5 + 0.866i)31-s + (−0.499 − 0.866i)35-s + 0.999·36-s + (0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (−0.499 + 0.866i)16-s + (0.5 + 0.866i)17-s + 0.999·20-s + (−0.5 + 0.866i)23-s + (−0.499 − 0.866i)25-s + 0.999·28-s − 29-s + (0.5 + 0.866i)31-s + (−0.499 − 0.866i)35-s + 0.999·36-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(805\)    =    \(5 \cdot 7 \cdot 23\)
Sign: $-0.0633 - 0.997i$
Analytic conductor: \(0.401747\)
Root analytic conductor: \(0.633835\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{805} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 805,\ (\ :0),\ -0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5799676889\)
\(L(\frac12)\) \(\approx\) \(0.5799676889\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - 2T + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.65705913530951020011236173738, −9.923418271362977624409477789329, −9.055271466228076740712977874285, −8.184280925680696503436258639093, −7.28348711280075040011178085596, −5.97376064887276166729870883827, −5.70460251213427331310398540339, −4.38352028793275125433918324314, −3.21181343389304225950462441678, −2.02053767034071231581372505253, 0.60199641936803271566918039828, 2.95783931472892270323240932627, 3.90163027072443659515872404211, 4.54321097389857710952685343794, 5.79785604015513397725085641841, 6.99221294721397798447025718931, 7.77993177621215237584568279878, 8.523893275148418110540594364639, 9.359167044997580716634310273867, 9.942339981521415992471320759637

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