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Properties

Label 2-799-799.234-c0-0-4
Degree 22
Conductor 799799
Sign −0.939−0.341i-0.939 - 0.341i
Analytic cond. 0.3987520.398752
Root an. cond. 0.6314680.631468
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61i·2-s + (−0.642 − 0.642i)3-s − 1.61·4-s + (−1.03 + 1.03i)6-s + (1.39 − 1.39i)7-s + i·8-s − 0.175i·9-s + (1.03 + 1.03i)12-s + (−2.26 − 2.26i)14-s + (0.309 + 0.951i)17-s − 0.284·18-s − 1.79·21-s + (0.642 − 0.642i)24-s + i·25-s + (−0.754 + 0.754i)27-s + (−2.26 + 2.26i)28-s + ⋯
L(s)  = 1  − 1.61i·2-s + (−0.642 − 0.642i)3-s − 1.61·4-s + (−1.03 + 1.03i)6-s + (1.39 − 1.39i)7-s + i·8-s − 0.175i·9-s + (1.03 + 1.03i)12-s + (−2.26 − 2.26i)14-s + (0.309 + 0.951i)17-s − 0.284·18-s − 1.79·21-s + (0.642 − 0.642i)24-s + i·25-s + (−0.754 + 0.754i)27-s + (−2.26 + 2.26i)28-s + ⋯

Functional equation

Λ(s)=(799s/2 ΓC(s) L(s)=((−0.939−0.341i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(799s/2 ΓC(s) L(s)=((−0.939−0.341i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 799 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 799799    =    17⋅4717 \cdot 47
Sign: −0.939−0.341i-0.939 - 0.341i
Analytic conductor: 0.3987520.398752
Root analytic conductor: 0.6314680.631468
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ799(234,⋅)\chi_{799} (234, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 799, ( :0), −0.939−0.341i)(2,\ 799,\ (\ :0),\ -0.939 - 0.341i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.82825381860.8282538186
L(12)L(\frac12) ≈\approx 0.82825381860.8282538186
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad17 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
47 1+T 1 + T
good2 1+1.61iT−T2 1 + 1.61iT - T^{2}
3 1+(0.642+0.642i)T+iT2 1 + (0.642 + 0.642i)T + iT^{2}
5 1−iT2 1 - iT^{2}
7 1+(−1.39+1.39i)T−iT2 1 + (-1.39 + 1.39i)T - iT^{2}
11 1+iT2 1 + iT^{2}
13 1−T2 1 - T^{2}
19 1+T2 1 + T^{2}
23 1+iT2 1 + iT^{2}
29 1−iT2 1 - iT^{2}
31 1−iT2 1 - iT^{2}
37 1+(−0.642−0.642i)T+iT2 1 + (-0.642 - 0.642i)T + iT^{2}
41 1+iT2 1 + iT^{2}
43 1+T2 1 + T^{2}
53 1−1.17iT−T2 1 - 1.17iT - T^{2}
59 1+1.61iT−T2 1 + 1.61iT - T^{2}
61 1+(0.221−0.221i)T−iT2 1 + (0.221 - 0.221i)T - iT^{2}
67 1−T2 1 - T^{2}
71 1+(−1.26−1.26i)T+iT2 1 + (-1.26 - 1.26i)T + iT^{2}
73 1−iT2 1 - iT^{2}
79 1+(−1.26+1.26i)T−iT2 1 + (-1.26 + 1.26i)T - iT^{2}
83 1−T2 1 - T^{2}
89 1+1.61T+T2 1 + 1.61T + T^{2}
97 1+(−0.221−0.221i)T+iT2 1 + (-0.221 - 0.221i)T + iT^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.36276273234322013119571749242, −9.546023718494355919555346824137, −8.356620478294143936276681588118, −7.52461668341903366279125653933, −6.56700279282551973566623634428, −5.22754030500683862527808367035, −4.27425796892834831771647267274, −3.45622213338103050020346655998, −1.76000343888954358364343851002, −1.06632027335765596283969352211, 2.33726233986095704338060907860, 4.39969481491432142859437293153, 5.08973779805699532604407337442, 5.53987265476286844157492164347, 6.40923652133995615017580660110, 7.64335380440630616265623353621, 8.195149374214521307408596146706, 8.977167894973261517257512384732, 9.828090828557289385108406115400, 11.06532285430356677351376511829

Graph of the ZZ-function along the critical line