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Properties

Label 2-650-5.4-c5-0-8
Degree 22
Conductor 650650
Sign −0.447+0.894i-0.447 + 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 + 19.2i·3-s − 16·4-s − 76.8·6-s − 51.1i·7-s − 64i·8-s − 126.·9-s + 494.·11-s − 307. i·12-s + 169i·13-s + 204.·14-s + 256·16-s − 2.36e3i·17-s − 506. i·18-s + 699.·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.23i·3-s − 0.5·4-s − 0.872·6-s − 0.394i·7-s − 0.353i·8-s − 0.520·9-s + 1.23·11-s − 0.616i·12-s + 0.277i·13-s + 0.279·14-s + 0.250·16-s − 1.98i·17-s − 0.368i·18-s + 0.444·19-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+0.894i-0.447 + 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 0.82281828370.8228182837
L(12)L(\frac12) ≈\approx 0.82281828370.8228182837
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 1−19.2iT−243T2 1 - 19.2iT - 243T^{2}
7 1+51.1iT−1.68e4T2 1 + 51.1iT - 1.68e4T^{2}
11 1−494.T+1.61e5T2 1 - 494.T + 1.61e5T^{2}
17 1+2.36e3iT−1.41e6T2 1 + 2.36e3iT - 1.41e6T^{2}
19 1−699.T+2.47e6T2 1 - 699.T + 2.47e6T^{2}
23 1−3.80e3iT−6.43e6T2 1 - 3.80e3iT - 6.43e6T^{2}
29 1+2.26e3T+2.05e7T2 1 + 2.26e3T + 2.05e7T^{2}
31 1+9.08e3T+2.86e7T2 1 + 9.08e3T + 2.86e7T^{2}
37 1+4.48e3iT−6.93e7T2 1 + 4.48e3iT - 6.93e7T^{2}
41 1+1.24e4T+1.15e8T2 1 + 1.24e4T + 1.15e8T^{2}
43 1−1.25e4iT−1.47e8T2 1 - 1.25e4iT - 1.47e8T^{2}
47 1−3.60e3iT−2.29e8T2 1 - 3.60e3iT - 2.29e8T^{2}
53 1+7.54e3iT−4.18e8T2 1 + 7.54e3iT - 4.18e8T^{2}
59 1+5.00e4T+7.14e8T2 1 + 5.00e4T + 7.14e8T^{2}
61 1+2.47e4T+8.44e8T2 1 + 2.47e4T + 8.44e8T^{2}
67 1−4.37e4iT−1.35e9T2 1 - 4.37e4iT - 1.35e9T^{2}
71 1−7.60e3T+1.80e9T2 1 - 7.60e3T + 1.80e9T^{2}
73 1−2.16e4iT−2.07e9T2 1 - 2.16e4iT - 2.07e9T^{2}
79 1+1.03e5T+3.07e9T2 1 + 1.03e5T + 3.07e9T^{2}
83 1−8.03e4iT−3.93e9T2 1 - 8.03e4iT - 3.93e9T^{2}
89 1+5.93e4T+5.58e9T2 1 + 5.93e4T + 5.58e9T^{2}
97 1+7.34e4iT−8.58e9T2 1 + 7.34e4iT - 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

−10.03594608384013982426297663478, −9.281568845852005915115115507881, −9.100477367289480306613518358409, −7.49701310697984778208323927726, −7.01463455501888753939298579420, −5.69599434135675158464679069133, −4.91076561495714424923936568113, −4.02481829338212262406214068930, −3.27596891446237640375953424171, −1.33035435032325176047832885419, 0.17792521344228198541589631241, 1.45490348744143616148447274799, 1.93858970345071507794288773128, 3.32659167346692373838370124474, 4.31966823979743706088062782619, 5.77644640849140991196864230450, 6.47624647882894459242544956580, 7.45490496960687785093462692422, 8.456175823487053119890965243047, 9.028335535635023629845334942652

Graph of the ZZ-function along the critical line