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Properties

Label 2-650-5.4-c5-0-69
Degree 22
Conductor 650650
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 104.249104.249
Root an. cond. 10.210210.2102
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4i·2-s − 16·4-s + 170i·7-s − 64i·8-s + 243·9-s − 250·11-s − 169i·13-s − 680·14-s + 256·16-s − 1.06e3i·17-s + 972i·18-s + 78·19-s − 1.00e3i·22-s + 1.57e3i·23-s + 676·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 1.31i·7-s − 0.353i·8-s + 9-s − 0.622·11-s − 0.277i·13-s − 0.927·14-s + 0.250·16-s − 0.891i·17-s + 0.707i·18-s + 0.0495·19-s − 0.440i·22-s + 0.621i·23-s + 0.196·26-s + ⋯

Functional equation

Λ(s)=(650s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(650s/2 ΓC(s+5/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 650650    =    2⋅52⋅132 \cdot 5^{2} \cdot 13
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 104.249104.249
Root analytic conductor: 10.210210.2102
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ650(599,⋅)\chi_{650} (599, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 650, ( :5/2), 0.447+0.894i)(2,\ 650,\ (\ :5/2),\ 0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 0.61007559930.6100755993
L(12)L(\frac12) ≈\approx 0.61007559930.6100755993
L(72)L(\frac{7}{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 1−4iT 1 - 4iT
5 1 1
13 1+169iT 1 + 169iT
good3 1−243T2 1 - 243T^{2}
7 1−170iT−1.68e4T2 1 - 170iT - 1.68e4T^{2}
11 1+250T+1.61e5T2 1 + 250T + 1.61e5T^{2}
17 1+1.06e3iT−1.41e6T2 1 + 1.06e3iT - 1.41e6T^{2}
19 1−78T+2.47e6T2 1 - 78T + 2.47e6T^{2}
23 1−1.57e3iT−6.43e6T2 1 - 1.57e3iT - 6.43e6T^{2}
29 1+2.57e3T+2.05e7T2 1 + 2.57e3T + 2.05e7T^{2}
31 1+8.65e3T+2.86e7T2 1 + 8.65e3T + 2.86e7T^{2}
37 1+1.09e4iT−6.93e7T2 1 + 1.09e4iT - 6.93e7T^{2}
41 1−1.05e3T+1.15e8T2 1 - 1.05e3T + 1.15e8T^{2}
43 1+5.90e3iT−1.47e8T2 1 + 5.90e3iT - 1.47e8T^{2}
47 1−5.96e3iT−2.29e8T2 1 - 5.96e3iT - 2.29e8T^{2}
53 1−2.90e4iT−4.18e8T2 1 - 2.90e4iT - 4.18e8T^{2}
59 1−1.39e4T+7.14e8T2 1 - 1.39e4T + 7.14e8T^{2}
61 1+3.28e4T+8.44e8T2 1 + 3.28e4T + 8.44e8T^{2}
67 1−6.95e4iT−1.35e9T2 1 - 6.95e4iT - 1.35e9T^{2}
71 1+5.05e4T+1.80e9T2 1 + 5.05e4T + 1.80e9T^{2}
73 1+4.67e4iT−2.07e9T2 1 + 4.67e4iT - 2.07e9T^{2}
79 1−1.93e4T+3.07e9T2 1 - 1.93e4T + 3.07e9T^{2}
83 1+8.74e4iT−3.93e9T2 1 + 8.74e4iT - 3.93e9T^{2}
89 1+9.41e4T+5.58e9T2 1 + 9.41e4T + 5.58e9T^{2}
97 1+1.82e5iT−8.58e9T2 1 + 1.82e5iT - 8.58e9T^{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

−9.338332789163391110963464133790, −8.881311486413362419027762561393, −7.62842051931812832181152162594, −7.19458845854385260263934825074, −5.81876445239945086251291291849, −5.36825008280670522608614138551, −4.24330809710752194713377085436, −2.94276659651090590896011864796, −1.73768026712978313587296239269, −0.13907298106046146538449847430, 1.06935338525208003745537233715, 1.99381650084970716669695554880, 3.48944317235194583476508427491, 4.18269143092090988590227267580, 5.09010932719553827065645751240, 6.49492079448778504743548669428, 7.38754431193185287615937658385, 8.156622463232810170321130873854, 9.327414086266412821706191588042, 10.20342167150840282451778053937

Graph of the ZZ-function along the critical line