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Properties

Label 2-799-799.798-c0-0-4
Degree $2$
Conductor $799$
Sign $1$
Analytic cond. $0.398752$
Root an. cond. $0.631468$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 1.41·5-s + 9-s − 1.41·11-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·29-s − 1.41·31-s − 36-s + 1.41·41-s + 1.41·44-s + 1.41·45-s + 47-s + 49-s − 2·53-s − 2.00·55-s − 64-s − 68-s − 1.41·73-s + 1.41·80-s + 81-s − 2·83-s + 1.41·85-s − 2·89-s + ⋯
L(s)  = 1  − 4-s + 1.41·5-s + 9-s − 1.41·11-s + 16-s + 17-s − 1.41·20-s + 1.41·23-s + 1.00·25-s − 1.41·29-s − 1.41·31-s − 36-s + 1.41·41-s + 1.41·44-s + 1.41·45-s + 47-s + 49-s − 2·53-s − 2.00·55-s − 64-s − 68-s − 1.41·73-s + 1.41·80-s + 81-s − 2·83-s + 1.41·85-s − 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $1$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (798, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.012569344\)
\(L(\frac12)\) \(\approx\) \(1.012569344\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 - T \)
47 \( 1 - T \)
good2 \( 1 + T^{2} \)
3 \( 1 - T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 - T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.33348361815926683397981824937, −9.525512684286570793824743923626, −9.153165933436031485356074601632, −7.87659228110464969574719270538, −7.15680822313243672563890265455, −5.60388819097852932686624349106, −5.42920296113198987194662399730, −4.23653969168912101210015572507, −2.89506002251114788168038342748, −1.48972277724596277388070807012, 1.48972277724596277388070807012, 2.89506002251114788168038342748, 4.23653969168912101210015572507, 5.42920296113198987194662399730, 5.60388819097852932686624349106, 7.15680822313243672563890265455, 7.87659228110464969574719270538, 9.153165933436031485356074601632, 9.525512684286570793824743923626, 10.33348361815926683397981824937

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