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Properties

Label 2-799-799.610-c0-0-4
Degree $2$
Conductor $799$
Sign $0.235 + 0.971i$
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  + (0.831 + 0.831i)2-s + (−0.581 − 1.40i)3-s + 0.381i·4-s + (0.684 − 1.65i)6-s + (−1.57 − 0.652i)7-s + (0.513 − 0.513i)8-s + (−0.928 + 0.928i)9-s + (0.536 − 0.222i)12-s + (−0.767 − 1.85i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 1.54·18-s + 2.59i·21-s + (−1.02 − 0.422i)24-s + (0.707 − 0.707i)25-s + ⋯
L(s)  = 1  + (0.831 + 0.831i)2-s + (−0.581 − 1.40i)3-s + 0.381i·4-s + (0.684 − 1.65i)6-s + (−1.57 − 0.652i)7-s + (0.513 − 0.513i)8-s + (−0.928 + 0.928i)9-s + (0.536 − 0.222i)12-s + (−0.767 − 1.85i)14-s + 1.23·16-s + (−0.309 − 0.951i)17-s − 1.54·18-s + 2.59i·21-s + (−1.02 − 0.422i)24-s + (0.707 − 0.707i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(799\)    =    \(17 \cdot 47\)
Sign: $0.235 + 0.971i$
Analytic conductor: \(0.398752\)
Root analytic conductor: \(0.631468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{799} (610, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 799,\ (\ :0),\ 0.235 + 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.020000971\)
\(L(\frac12)\) \(\approx\) \(1.020000971\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 - iT \)
good2 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
3 \( 1 + (0.581 + 1.40i)T + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (1.57 + 0.652i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (-0.497 - 1.20i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
53 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
59 \( 1 + (-1.14 + 1.14i)T - iT^{2} \)
61 \( 1 + (-1.79 - 0.744i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.763 - 1.84i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.763 + 1.84i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 - 1.17iT - T^{2} \)
97 \( 1 + (0.431 - 0.178i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.28334715799356289666740404040, −9.537028490037877585406052736239, −8.123786997797798011474079398576, −7.10318596731421927655803276581, −6.77534173904016297911675028650, −6.19942502306389733751434361438, −5.26714025412207548659826200789, −4.08830200386820838153504094096, −2.78373412992047753801095905578, −0.894786694864230422643487387847, 2.46756349797084567514259080872, 3.54773911631690299192247486021, 4.00886176312839353012338363869, 5.20123193370345540502669099253, 5.79183669810009959197487910181, 6.87091329499942075002468297117, 8.452694060189380969556852050973, 9.357777363898797653919917815141, 10.01966088943660178773291077865, 10.75556802865747950595435896802

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