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Properties

Label 2-10-5.2-c4-0-1
Degree 22
Conductor 1010
Sign 0.640+0.767i0.640 + 0.767i
Analytic cond. 1.033691.03369
Root an. cond. 1.016711.01671
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 − 2i)2-s + (9 − 9i)3-s + 8i·4-s + (−15 + 20i)5-s − 36·6-s + (29 + 29i)7-s + (16 − 16i)8-s − 81i·9-s + (70 − 10i)10-s − 118·11-s + (72 + 72i)12-s + (69 − 69i)13-s − 116i·14-s + (45 + 315i)15-s − 64·16-s + (−271 − 271i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (1 − i)3-s + 0.5i·4-s + (−0.599 + 0.800i)5-s − 6-s + (0.591 + 0.591i)7-s + (0.250 − 0.250i)8-s − i·9-s + (0.700 − 0.100i)10-s − 0.975·11-s + (0.5 + 0.5i)12-s + (0.408 − 0.408i)13-s − 0.591i·14-s + (0.200 + 1.39i)15-s − 0.250·16-s + (−0.937 − 0.937i)17-s + ⋯

Functional equation

Λ(s)=(10s/2 ΓC(s) L(s)=((0.640+0.767i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(10s/2 ΓC(s+2) L(s)=((0.640+0.767i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 1010    =    2⋅52 \cdot 5
Sign: 0.640+0.767i0.640 + 0.767i
Analytic conductor: 1.033691.03369
Root analytic conductor: 1.016711.01671
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ10(7,⋅)\chi_{10} (7, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 10, ( :2), 0.640+0.767i)(2,\ 10,\ (\ :2),\ 0.640 + 0.767i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.905951−0.423920i0.905951 - 0.423920i
L(12)L(\frac12) ≈\approx 0.905951−0.423920i0.905951 - 0.423920i
L(3)L(3) 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)
bad2 1+(2+2i)T 1 + (2 + 2i)T
5 1+(15−20i)T 1 + (15 - 20i)T
good3 1+(−9+9i)T−81iT2 1 + (-9 + 9i)T - 81iT^{2}
7 1+(−29−29i)T+2.40e3iT2 1 + (-29 - 29i)T + 2.40e3iT^{2}
11 1+118T+1.46e4T2 1 + 118T + 1.46e4T^{2}
13 1+(−69+69i)T−2.85e4iT2 1 + (-69 + 69i)T - 2.85e4iT^{2}
17 1+(271+271i)T+8.35e4iT2 1 + (271 + 271i)T + 8.35e4iT^{2}
19 1−280iT−1.30e5T2 1 - 280iT - 1.30e5T^{2}
23 1+(−269+269i)T−2.79e5iT2 1 + (-269 + 269i)T - 2.79e5iT^{2}
29 1−680iT−7.07e5T2 1 - 680iT - 7.07e5T^{2}
31 1−202T+9.23e5T2 1 - 202T + 9.23e5T^{2}
37 1+(651+651i)T+1.87e6iT2 1 + (651 + 651i)T + 1.87e6iT^{2}
41 1−1.68e3T+2.82e6T2 1 - 1.68e3T + 2.82e6T^{2}
43 1+(−1.08e3+1.08e3i)T−3.41e6iT2 1 + (-1.08e3 + 1.08e3i)T - 3.41e6iT^{2}
47 1+(−1.26e3−1.26e3i)T+4.87e6iT2 1 + (-1.26e3 - 1.26e3i)T + 4.87e6iT^{2}
53 1+(611−611i)T−7.89e6iT2 1 + (611 - 611i)T - 7.89e6iT^{2}
59 1−1.16e3iT−1.21e7T2 1 - 1.16e3iT - 1.21e7T^{2}
61 1+5.59e3T+1.38e7T2 1 + 5.59e3T + 1.38e7T^{2}
67 1+(751+751i)T+2.01e7iT2 1 + (751 + 751i)T + 2.01e7iT^{2}
71 1−6.44e3T+2.54e7T2 1 - 6.44e3T + 2.54e7T^{2}
73 1+(2.95e3−2.95e3i)T−2.83e7iT2 1 + (2.95e3 - 2.95e3i)T - 2.83e7iT^{2}
79 1−1.05e4iT−3.89e7T2 1 - 1.05e4iT - 3.89e7T^{2}
83 1+(6.23e3−6.23e3i)T−4.74e7iT2 1 + (6.23e3 - 6.23e3i)T - 4.74e7iT^{2}
89 1+1.44e4iT−6.27e7T2 1 + 1.44e4iT - 6.27e7T^{2}
97 1+(7.31e3+7.31e3i)T+8.85e7iT2 1 + (7.31e3 + 7.31e3i)T + 8.85e7iT^{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

−19.83766310813537809024435622667, −18.57683806843483705103387378519, −18.17528192393846041571962560345, −15.58359723051238049094321415300, −14.09689758628053902298710611143, −12.56296324811360941388841825951, −10.92683241127515175220492978638, −8.576419223140330273316556010163, −7.41236298354186449641074172392, −2.63299884198847838915051193532, 4.46846646297317084744423291953, 7.961397833103900235480438224705, 9.104051683033048217187035624946, 10.84596160869241973199237549649, 13.50900885747239028179362041665, 15.14653380220326052994219103775, 15.92584702397339250974258949644, 17.39657263892531068990599874961, 19.37004415030012598391470815480, 20.40927505534607014480708864434

Graph of the ZZ-function along the critical line