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Properties

Label 2-1050-5.4-c3-0-30
Degree 22
Conductor 10501050
Sign 0.894−0.447i0.894 - 0.447i
Analytic cond. 61.952061.9520
Root an. cond. 7.870957.87095
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 3i·3-s − 4·4-s − 6·6-s − 7i·7-s − 8i·8-s − 9·9-s − 12i·12-s + 26i·13-s + 14·14-s + 16·16-s − 18i·17-s − 18i·18-s − 92·19-s + 21·21-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.377i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s + 0.554i·13-s + 0.267·14-s + 0.250·16-s − 0.256i·17-s − 0.235i·18-s − 1.11·19-s + 0.218·21-s + ⋯

Functional equation

Λ(s)=(1050s/2 ΓC(s) L(s)=((0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(1050s/2 ΓC(s+3/2) L(s)=((0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10501050    =    2⋅3⋅52⋅72 \cdot 3 \cdot 5^{2} \cdot 7
Sign: 0.894−0.447i0.894 - 0.447i
Analytic conductor: 61.952061.9520
Root analytic conductor: 7.870957.87095
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ1050(799,⋅)\chi_{1050} (799, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1050, ( :3/2), 0.894−0.447i)(2,\ 1050,\ (\ :3/2),\ 0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.4792424601.479242460
L(12)L(\frac12) ≈\approx 1.4792424601.479242460
L(52)L(\frac{5}{2}) 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 1−2iT 1 - 2iT
3 1−3iT 1 - 3iT
5 1 1
7 1+7iT 1 + 7iT
good11 1+1.33e3T2 1 + 1.33e3T^{2}
13 1−26iT−2.19e3T2 1 - 26iT - 2.19e3T^{2}
17 1+18iT−4.91e3T2 1 + 18iT - 4.91e3T^{2}
19 1+92T+6.85e3T2 1 + 92T + 6.85e3T^{2}
23 1−1.21e4T2 1 - 1.21e4T^{2}
29 1−6T+2.43e4T2 1 - 6T + 2.43e4T^{2}
31 1+4T+2.97e4T2 1 + 4T + 2.97e4T^{2}
37 1+410iT−5.06e4T2 1 + 410iT - 5.06e4T^{2}
41 1−174T+6.89e4T2 1 - 174T + 6.89e4T^{2}
43 1−248iT−7.95e4T2 1 - 248iT - 7.95e4T^{2}
47 1+420iT−1.03e5T2 1 + 420iT - 1.03e5T^{2}
53 1−102iT−1.48e5T2 1 - 102iT - 1.48e5T^{2}
59 1−588T+2.05e5T2 1 - 588T + 2.05e5T^{2}
61 1−650T+2.26e5T2 1 - 650T + 2.26e5T^{2}
67 1+152iT−3.00e5T2 1 + 152iT - 3.00e5T^{2}
71 1+168T+3.57e5T2 1 + 168T + 3.57e5T^{2}
73 1+610iT−3.89e5T2 1 + 610iT - 3.89e5T^{2}
79 1−1.04e3T+4.93e5T2 1 - 1.04e3T + 4.93e5T^{2}
83 1+684iT−5.71e5T2 1 + 684iT - 5.71e5T^{2}
89 1−834T+7.04e5T2 1 - 834T + 7.04e5T^{2}
97 1+110iT−9.12e5T2 1 + 110iT - 9.12e5T^{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

−9.428712119352182343826677400094, −8.813300547246414527212662215880, −7.928099136960535259483017847649, −7.03315385832818460396791929396, −6.24152549020998964330523329032, −5.28772071252497435373384869198, −4.37025237511892437856892159129, −3.66979604113024556571203763163, −2.20024820603368514896279211342, −0.48559575863204253630972390157, 0.834702101896117258357679907546, 2.02399222979672837627205515814, 2.90268356798282158583098580144, 4.02711178236988241265120013929, 5.12325342945012997279435123916, 6.04256384218273206046609311524, 6.92428786950056481267015988856, 8.100363265798700095998177848210, 8.543508375563387912046061180746, 9.556963018865082973074995203754

Graph of the ZZ-function along the critical line