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Properties

Label 2-650-5.4-c5-0-27
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 − 3.71i·3-s − 16·4-s + 14.8·6-s + 10.2i·7-s − 64i·8-s + 229.·9-s + 197.·11-s + 59.4i·12-s + 169i·13-s − 41.0·14-s + 256·16-s + 949. i·17-s + 916. i·18-s − 2.23e3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.238i·3-s − 0.5·4-s + 0.168·6-s + 0.0791i·7-s − 0.353i·8-s + 0.943·9-s + 0.491·11-s + 0.119i·12-s + 0.277i·13-s − 0.0559·14-s + 0.250·16-s + 0.797i·17-s + 0.666i·18-s − 1.41·19-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 1.8039417681.803941768
L(12)L(\frac12) ≈\approx 1.8039417681.803941768
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+3.71iT−243T2 1 + 3.71iT - 243T^{2}
7 1−10.2iT−1.68e4T2 1 - 10.2iT - 1.68e4T^{2}
11 1−197.T+1.61e5T2 1 - 197.T + 1.61e5T^{2}
17 1−949.iT−1.41e6T2 1 - 949. iT - 1.41e6T^{2}
19 1+2.23e3T+2.47e6T2 1 + 2.23e3T + 2.47e6T^{2}
23 1+367.iT−6.43e6T2 1 + 367. iT - 6.43e6T^{2}
29 1−6.60e3T+2.05e7T2 1 - 6.60e3T + 2.05e7T^{2}
31 1+9.91e3T+2.86e7T2 1 + 9.91e3T + 2.86e7T^{2}
37 1−9.39e3iT−6.93e7T2 1 - 9.39e3iT - 6.93e7T^{2}
41 1−2.05e4T+1.15e8T2 1 - 2.05e4T + 1.15e8T^{2}
43 1+1.63e4iT−1.47e8T2 1 + 1.63e4iT - 1.47e8T^{2}
47 1−1.35e4iT−2.29e8T2 1 - 1.35e4iT - 2.29e8T^{2}
53 1+3.50e4iT−4.18e8T2 1 + 3.50e4iT - 4.18e8T^{2}
59 1−6.17e3T+7.14e8T2 1 - 6.17e3T + 7.14e8T^{2}
61 1−1.37e4T+8.44e8T2 1 - 1.37e4T + 8.44e8T^{2}
67 1−6.21e4iT−1.35e9T2 1 - 6.21e4iT - 1.35e9T^{2}
71 1+3.83e4T+1.80e9T2 1 + 3.83e4T + 1.80e9T^{2}
73 1−5.99e4iT−2.07e9T2 1 - 5.99e4iT - 2.07e9T^{2}
79 1+8.96e4T+3.07e9T2 1 + 8.96e4T + 3.07e9T^{2}
83 1+3.81e4iT−3.93e9T2 1 + 3.81e4iT - 3.93e9T^{2}
89 1−7.82e4T+5.58e9T2 1 - 7.82e4T + 5.58e9T^{2}
97 1+8.05e4iT−8.58e9T2 1 + 8.05e4iT - 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

−10.01490965016339175430817571516, −8.963840970674933210702280936329, −8.327843018468920554763025387653, −7.25022698745281790057090244978, −6.61958254984071574936630875872, −5.77717998378022361716435020857, −4.49463412409470356696621570470, −3.86125847059985752723520927594, −2.17883960659460888311907623577, −1.02559934351973872989535480549, 0.44409259208751514494333250563, 1.58768049422526518156796549212, 2.71539693346077122485768156496, 3.96658780277569106644300777851, 4.54960975823399968666830598478, 5.76425401034089152965326611079, 6.90333102816494350149405095376, 7.79426425534590172251355276095, 8.994315736877392583497952689137, 9.484984370860072076282322543736

Graph of the ZZ-function along the critical line