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Properties

Label 2-799-799.281-c0-0-3
Degree $2$
Conductor $799$
Sign $0.673 - 0.739i$
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  + (1.70 + 0.707i)3-s + i·4-s + (−0.707 − 1.70i)7-s + (1.70 + 1.70i)9-s + (−0.707 + 1.70i)12-s − 16-s − 17-s − 3.41i·21-s + (−0.707 − 0.707i)25-s + (1 + 2.41i)27-s + (1.70 − 0.707i)28-s + (−1.70 + 1.70i)36-s + (−0.707 − 0.292i)37-s − i·47-s + (−1.70 − 0.707i)48-s + (−1.70 + 1.70i)49-s + ⋯
L(s)  = 1  + (1.70 + 0.707i)3-s + i·4-s + (−0.707 − 1.70i)7-s + (1.70 + 1.70i)9-s + (−0.707 + 1.70i)12-s − 16-s − 17-s − 3.41i·21-s + (−0.707 − 0.707i)25-s + (1 + 2.41i)27-s + (1.70 − 0.707i)28-s + (−1.70 + 1.70i)36-s + (−0.707 − 0.292i)37-s − i·47-s + (−1.70 − 0.707i)48-s + (−1.70 + 1.70i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.673 - 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.673 - 0.739i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.673 - 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.509704185\)
\(L(\frac12)\) \(\approx\) \(1.509704185\)
\(L(1)\) 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 \( 1 + iT \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + (-0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 - iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.707 + 0.707i)T^{2} \)
79 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
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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

−10.32909644719488401316218463218, −9.658515791994381735266104072996, −8.844141940531045314116189394852, −8.098161144334781831337989042563, −7.37889227873632408428905342381, −6.72170602454147341278508062384, −4.56935911595939851380123199415, −3.94593363704164899158115402591, −3.35970391375422249321562918347, −2.27441389998862881456962322526, 1.84847062897443005881527195053, 2.49529307446799026054557928504, 3.57063600973940416759050567102, 5.08739090476776641076545251497, 6.23078548815443055948851362510, 6.80543112905942585348483167052, 8.033295387806848099194758465700, 8.830710144917176095301165319054, 9.337966373836278716840627630196, 9.868450587903769788551304139406

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