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Properties

Label 2-2100-5.4-c3-0-15
Degree 22
Conductor 21002100
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 123.904123.904
Root an. cond. 11.131211.1312
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  + 3i·3-s + 7i·7-s − 9·9-s + 6.11·11-s + 15.2i·13-s + 101. i·17-s + 72.9·19-s − 21·21-s + 138. i·23-s − 27i·27-s − 71.7·29-s + 212.·31-s + 18.3i·33-s − 66.9i·37-s − 45.6·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 0.167·11-s + 0.324i·13-s + 1.44i·17-s + 0.880·19-s − 0.218·21-s + 1.25i·23-s − 0.192i·27-s − 0.459·29-s + 1.23·31-s + 0.0968i·33-s − 0.297i·37-s − 0.187·39-s + ⋯

Functional equation

Λ(s)=(2100s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(2100s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{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: 21002100    =    22⋅3⋅52⋅72^{2} \cdot 3 \cdot 5^{2} \cdot 7
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 123.904123.904
Root analytic conductor: 11.131211.1312
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ2100(1849,⋅)\chi_{2100} (1849, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2100, ( :3/2), −0.894−0.447i)(2,\ 2100,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 1.6506955551.650695555
L(12)L(\frac12) ≈\approx 1.6506955551.650695555
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 1
3 1−3iT 1 - 3iT
5 1 1
7 1−7iT 1 - 7iT
good11 1−6.11T+1.33e3T2 1 - 6.11T + 1.33e3T^{2}
13 1−15.2iT−2.19e3T2 1 - 15.2iT - 2.19e3T^{2}
17 1−101.iT−4.91e3T2 1 - 101. iT - 4.91e3T^{2}
19 1−72.9T+6.85e3T2 1 - 72.9T + 6.85e3T^{2}
23 1−138.iT−1.21e4T2 1 - 138. iT - 1.21e4T^{2}
29 1+71.7T+2.43e4T2 1 + 71.7T + 2.43e4T^{2}
31 1−212.T+2.97e4T2 1 - 212.T + 2.97e4T^{2}
37 1+66.9iT−5.06e4T2 1 + 66.9iT - 5.06e4T^{2}
41 1+12.4T+6.89e4T2 1 + 12.4T + 6.89e4T^{2}
43 1+398.iT−7.95e4T2 1 + 398. iT - 7.95e4T^{2}
47 1−176.iT−1.03e5T2 1 - 176. iT - 1.03e5T^{2}
53 1+131.iT−1.48e5T2 1 + 131. iT - 1.48e5T^{2}
59 1−654.T+2.05e5T2 1 - 654.T + 2.05e5T^{2}
61 1+120.T+2.26e5T2 1 + 120.T + 2.26e5T^{2}
67 1−310.iT−3.00e5T2 1 - 310. iT - 3.00e5T^{2}
71 1+400.T+3.57e5T2 1 + 400.T + 3.57e5T^{2}
73 1−243.iT−3.89e5T2 1 - 243. iT - 3.89e5T^{2}
79 1−553.T+4.93e5T2 1 - 553.T + 4.93e5T^{2}
83 1−756.iT−5.71e5T2 1 - 756. iT - 5.71e5T^{2}
89 1+80.1T+7.04e5T2 1 + 80.1T + 7.04e5T^{2}
97 1−1.10e3iT−9.12e5T2 1 - 1.10e3iT - 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

−9.115933993606922661109550739563, −8.411345360852994940456180229166, −7.62967474167803303135359313629, −6.66326967605597784807901054606, −5.78621672022298949313648431573, −5.18360490582439903557757144565, −4.07753635431654939132569353661, −3.47374845251050066080306907759, −2.31111213033764325647753014869, −1.21143135473428404046741468497, 0.37036535816253886462649550025, 1.18071549634752870099107060319, 2.49843957876396792451094506957, 3.22664717279421976539576011856, 4.45721565717215445250033683782, 5.19208010947653784634598832814, 6.20220851884038192561221006329, 6.91066186221949859099543521203, 7.60123775786496653545902087021, 8.291873611187854006842095537476

Graph of the ZZ-function along the critical line