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Properties

Label 2-650-5.4-c5-0-59
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 − 17.8i·3-s − 16·4-s + 71.2·6-s − 86.7i·7-s − 64i·8-s − 74.2·9-s − 203.·11-s + 284. i·12-s − 169i·13-s + 347.·14-s + 256·16-s + 406. i·17-s − 296. i·18-s + 2.32e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.14i·3-s − 0.5·4-s + 0.807·6-s − 0.669i·7-s − 0.353i·8-s − 0.305·9-s − 0.506·11-s + 0.571i·12-s − 0.277i·13-s + 0.473·14-s + 0.250·16-s + 0.341i·17-s − 0.216i·18-s + 1.47·19-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 2.0583656452.058365645
L(12)L(\frac12) ≈\approx 2.0583656452.058365645
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+17.8iT−243T2 1 + 17.8iT - 243T^{2}
7 1+86.7iT−1.68e4T2 1 + 86.7iT - 1.68e4T^{2}
11 1+203.T+1.61e5T2 1 + 203.T + 1.61e5T^{2}
17 1−406.iT−1.41e6T2 1 - 406. iT - 1.41e6T^{2}
19 1−2.32e3T+2.47e6T2 1 - 2.32e3T + 2.47e6T^{2}
23 1−3.28e3iT−6.43e6T2 1 - 3.28e3iT - 6.43e6T^{2}
29 1−2.72e3T+2.05e7T2 1 - 2.72e3T + 2.05e7T^{2}
31 1−4.91e3T+2.86e7T2 1 - 4.91e3T + 2.86e7T^{2}
37 1+3.12e3iT−6.93e7T2 1 + 3.12e3iT - 6.93e7T^{2}
41 1−1.50e4T+1.15e8T2 1 - 1.50e4T + 1.15e8T^{2}
43 1−2.79e3iT−1.47e8T2 1 - 2.79e3iT - 1.47e8T^{2}
47 1+1.66e4iT−2.29e8T2 1 + 1.66e4iT - 2.29e8T^{2}
53 1−1.72e4iT−4.18e8T2 1 - 1.72e4iT - 4.18e8T^{2}
59 1+2.76e3T+7.14e8T2 1 + 2.76e3T + 7.14e8T^{2}
61 1+1.68e4T+8.44e8T2 1 + 1.68e4T + 8.44e8T^{2}
67 1+1.90e4iT−1.35e9T2 1 + 1.90e4iT - 1.35e9T^{2}
71 1−8.32e3T+1.80e9T2 1 - 8.32e3T + 1.80e9T^{2}
73 1−4.94e4iT−2.07e9T2 1 - 4.94e4iT - 2.07e9T^{2}
79 1+9.58e4T+3.07e9T2 1 + 9.58e4T + 3.07e9T^{2}
83 1−3.41e4iT−3.93e9T2 1 - 3.41e4iT - 3.93e9T^{2}
89 1−2.66e4T+5.58e9T2 1 - 2.66e4T + 5.58e9T^{2}
97 1+2.49e4iT−8.58e9T2 1 + 2.49e4iT - 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

−9.572500880889281055573222555283, −8.387278002960991688276223787151, −7.51649438819996965801222050315, −7.28393007395354808916350403611, −6.17959569343121226102340625097, −5.35558898321617319849330852748, −4.14769519482778580216509904244, −2.88868773040646325603838085657, −1.40542989983443627680945384795, −0.58610135300322567807997016850, 0.913920297468290821391087662090, 2.45903098333016851064125675268, 3.24908430692276574402439896256, 4.44575963804332836497249691995, 5.01655846567430697260349276203, 6.06913689864061043906874800797, 7.46953300437329196435540208044, 8.579212389603211898334613537366, 9.305572352320784961918797905132, 9.990687047186043743232888909532

Graph of the ZZ-function along the critical line