Properties

Label 2-10e2-20.19-c10-0-47
Degree 22
Conductor 100100
Sign 0.951+0.308i-0.951 + 0.308i
Analytic cond. 63.535763.5357
Root an. cond. 7.970937.97093
Motivic weight 1010
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−20.8 − 24.2i)2-s − 208.·3-s + (−153. + 1.01e3i)4-s + (4.35e3 + 5.06e3i)6-s − 1.75e4·7-s + (2.77e4 − 1.74e4i)8-s − 1.54e4·9-s − 2.65e5i·11-s + (3.20e4 − 2.11e5i)12-s + 6.47e5i·13-s + (3.66e5 + 4.25e5i)14-s + (−1.00e6 − 3.10e5i)16-s + 2.51e6i·17-s + (3.22e5 + 3.75e5i)18-s − 1.70e5i·19-s + ⋯
L(s)  = 1  + (−0.652 − 0.758i)2-s − 0.859·3-s + (−0.149 + 0.988i)4-s + (0.560 + 0.651i)6-s − 1.04·7-s + (0.847 − 0.531i)8-s − 0.261·9-s − 1.65i·11-s + (0.128 − 0.849i)12-s + 1.74i·13-s + (0.681 + 0.792i)14-s + (−0.955 − 0.296i)16-s + 1.77i·17-s + (0.170 + 0.198i)18-s − 0.0688i·19-s + ⋯

Functional equation

Λ(s)=(100s/2ΓC(s)L(s)=((0.951+0.308i)Λ(11s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(11-s) \end{aligned}
Λ(s)=(100s/2ΓC(s+5)L(s)=((0.951+0.308i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 100100    =    22522^{2} \cdot 5^{2}
Sign: 0.951+0.308i-0.951 + 0.308i
Analytic conductor: 63.535763.5357
Root analytic conductor: 7.970937.97093
Motivic weight: 1010
Rational: no
Arithmetic: yes
Character: χ100(99,)\chi_{100} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 100, ( :5), 0.951+0.308i)(2,\ 100,\ (\ :5),\ -0.951 + 0.308i)

Particular Values

L(112)L(\frac{11}{2}) \approx 0.01567800.0992480i0.0156780 - 0.0992480i
L(12)L(\frac12) \approx 0.01567800.0992480i0.0156780 - 0.0992480i
L(6)L(6) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(20.8+24.2i)T 1 + (20.8 + 24.2i)T
5 1 1
good3 1+208.T+5.90e4T2 1 + 208.T + 5.90e4T^{2}
7 1+1.75e4T+2.82e8T2 1 + 1.75e4T + 2.82e8T^{2}
11 1+2.65e5iT2.59e10T2 1 + 2.65e5iT - 2.59e10T^{2}
13 16.47e5iT1.37e11T2 1 - 6.47e5iT - 1.37e11T^{2}
17 12.51e6iT2.01e12T2 1 - 2.51e6iT - 2.01e12T^{2}
19 1+1.70e5iT6.13e12T2 1 + 1.70e5iT - 6.13e12T^{2}
23 15.21e6T+4.14e13T2 1 - 5.21e6T + 4.14e13T^{2}
29 1+6.80e6T+4.20e14T2 1 + 6.80e6T + 4.20e14T^{2}
31 12.47e7iT8.19e14T2 1 - 2.47e7iT - 8.19e14T^{2}
37 1+9.23e6iT4.80e15T2 1 + 9.23e6iT - 4.80e15T^{2}
41 11.44e8T+1.34e16T2 1 - 1.44e8T + 1.34e16T^{2}
43 1+2.79e7T+2.16e16T2 1 + 2.79e7T + 2.16e16T^{2}
47 11.10e8T+5.25e16T2 1 - 1.10e8T + 5.25e16T^{2}
53 11.09e8iT1.74e17T2 1 - 1.09e8iT - 1.74e17T^{2}
59 16.65e8iT5.11e17T2 1 - 6.65e8iT - 5.11e17T^{2}
61 1+7.40e8T+7.13e17T2 1 + 7.40e8T + 7.13e17T^{2}
67 1+4.17e7T+1.82e18T2 1 + 4.17e7T + 1.82e18T^{2}
71 1+6.30e8iT3.25e18T2 1 + 6.30e8iT - 3.25e18T^{2}
73 19.52e8iT4.29e18T2 1 - 9.52e8iT - 4.29e18T^{2}
79 12.24e9iT9.46e18T2 1 - 2.24e9iT - 9.46e18T^{2}
83 1+6.38e9T+1.55e19T2 1 + 6.38e9T + 1.55e19T^{2}
89 11.29e9T+3.11e19T2 1 - 1.29e9T + 3.11e19T^{2}
97 1+1.43e10iT7.37e19T2 1 + 1.43e10iT - 7.37e19T^{2}
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   L(s)=p j=12(1αj,pps)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

−11.20393652381320073086051892260, −10.63198947850785595388545363534, −9.237950029131086090790535954593, −8.502421450229253341532755520252, −6.77936038409013764240768671651, −5.90078477300172406405398140521, −4.06890723768001349674530244596, −2.95464723455804753763753152347, −1.29576765126569838835528415877, −0.05649516821811321837979156607, 0.73772941933681314947828211822, 2.71742817233799169414248547913, 4.84764446641227093314048212432, 5.71149636782158546563360200385, 6.84037908965263089548195792766, 7.69643353718041627380086483711, 9.310078406246055415323722417317, 10.00297335199419996230311534865, 11.04333774392127372021252435305, 12.33563905934230534923541070250

Graph of the ZZ-function along the critical line