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Properties

Label 2-799-799.234-c0-0-3
Degree $2$
Conductor $799$
Sign $0.961 - 0.275i$
Analytic cond. $0.398752$
Root an. cond. $0.631468$
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  + 0.618i·2-s + (−0.221 − 0.221i)3-s + 0.618·4-s + (0.136 − 0.136i)6-s + (0.642 − 0.642i)7-s + i·8-s − 0.902i·9-s + (−0.136 − 0.136i)12-s + (0.396 + 0.396i)14-s + (−0.809 − 0.587i)17-s + 0.557·18-s − 0.284·21-s + (0.221 − 0.221i)24-s + i·25-s + (−0.420 + 0.420i)27-s + (0.396 − 0.396i)28-s + ⋯
L(s)  = 1  + 0.618i·2-s + (−0.221 − 0.221i)3-s + 0.618·4-s + (0.136 − 0.136i)6-s + (0.642 − 0.642i)7-s + i·8-s − 0.902i·9-s + (−0.136 − 0.136i)12-s + (0.396 + 0.396i)14-s + (−0.809 − 0.587i)17-s + 0.557·18-s − 0.284·21-s + (0.221 − 0.221i)24-s + i·25-s + (−0.420 + 0.420i)27-s + (0.396 − 0.396i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.961 - 0.275i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (234, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.961 - 0.275i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.125631769\)
\(L(\frac12)\) \(\approx\) \(1.125631769\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.809 + 0.587i)T \)
47 \( 1 + T \)
good2 \( 1 - 0.618iT - T^{2} \)
3 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
5 \( 1 - iT^{2} \)
7 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - 1.90iT - T^{2} \)
59 \( 1 - 0.618iT - T^{2} \)
61 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.618T + T^{2} \)
97 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
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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

−10.73444008164134370887114169788, −9.574257244015017322923054505536, −8.665242920766217113457990276462, −7.66824906152998017168633918940, −7.05680841141372253064926999143, −6.32585072265563079375187314641, −5.38267649122223876536397204972, −4.31041590630269292513213728810, −2.97073178173103081062108242035, −1.50021595385170185476727381398, 1.81603116458268595366090047535, 2.58582575141961639504201372891, 4.00011311388167499768079791635, 5.01079727752070267972009760957, 6.00162603757293823615768167264, 6.93869849391645339547969744342, 8.008839545527945066520295060311, 8.704147811499830345188998814028, 9.966639437166362958372919549371, 10.49338648165342073824202504039

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