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Properties

Label 2-950-19.11-c1-0-13
Degree 22
Conductor 950950
Sign 0.910+0.412i0.910 + 0.412i
Analytic cond. 7.585787.58578
Root an. cond. 2.754232.75423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 7-s − 0.999·8-s + (1.5 + 2.59i)9-s + 5·11-s + (1 + 1.73i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + 3·18-s + (−0.5 + 4.33i)19-s + (2.5 − 4.33i)22-s + (0.5 + 0.866i)23-s + 1.99·26-s + (0.499 + 0.866i)28-s + (−3 − 5.19i)29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.377·7-s − 0.353·8-s + (0.5 + 0.866i)9-s + 1.50·11-s + (0.277 + 0.480i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + 0.707·18-s + (−0.114 + 0.993i)19-s + (0.533 − 0.923i)22-s + (0.104 + 0.180i)23-s + 0.392·26-s + (0.0944 + 0.163i)28-s + (−0.557 − 0.964i)29-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((0.910+0.412i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((0.910+0.412i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 0.910+0.412i0.910 + 0.412i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(201,⋅)\chi_{950} (201, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 0.910+0.412i)(2,\ 950,\ (\ :1/2),\ 0.910 + 0.412i)

Particular Values

L(1)L(1) ≈\approx 1.94884−0.421198i1.94884 - 0.421198i
L(12)L(\frac12) ≈\approx 1.94884−0.421198i1.94884 - 0.421198i
L(32)L(\frac{3}{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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1 1
19 1+(0.5−4.33i)T 1 + (0.5 - 4.33i)T
good3 1+(−1.5−2.59i)T2 1 + (-1.5 - 2.59i)T^{2}
7 1+T+7T2 1 + T + 7T^{2}
11 1−5T+11T2 1 - 5T + 11T^{2}
13 1+(−1−1.73i)T+(−6.5+11.2i)T2 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2}
17 1+(−8.5−14.7i)T2 1 + (-8.5 - 14.7i)T^{2}
23 1+(−0.5−0.866i)T+(−11.5+19.9i)T2 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2}
29 1+(3+5.19i)T+(−14.5+25.1i)T2 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1−11T+37T2 1 - 11T + 37T^{2}
41 1+(−4.5+7.79i)T+(−20.5−35.5i)T2 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2}
43 1+(−3+5.19i)T+(−21.5−37.2i)T2 1 + (-3 + 5.19i)T + (-21.5 - 37.2i)T^{2}
47 1+(−23.5+40.7i)T2 1 + (-23.5 + 40.7i)T^{2}
53 1+(−2.5−4.33i)T+(−26.5+45.8i)T2 1 + (-2.5 - 4.33i)T + (-26.5 + 45.8i)T^{2}
59 1+(−29.5−51.0i)T2 1 + (-29.5 - 51.0i)T^{2}
61 1+(−30.5+52.8i)T2 1 + (-30.5 + 52.8i)T^{2}
67 1+(−6−10.3i)T+(−33.5+58.0i)T2 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2}
71 1+(3−5.19i)T+(−35.5−61.4i)T2 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2}
73 1+(−7+12.1i)T+(−36.5−63.2i)T2 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2}
79 1+(5−8.66i)T+(−39.5−68.4i)T2 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2}
83 1+14T+83T2 1 + 14T + 83T^{2}
89 1+(−3.5−6.06i)T+(−44.5+77.0i)T2 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2}
97 1+(1−1.73i)T+(−48.5−84.0i)T2 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{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

−9.932918427995837414089519391319, −9.430729454746527553327742374237, −8.447918354250738830884132284406, −7.40655059846664527146659403522, −6.41831436480062518115554336333, −5.64848032313846357543304691273, −4.29353267796000027187381777841, −3.87614472238841220322712760618, −2.40931606491552331823877825361, −1.29793947836325581910441911715, 1.04987745002552233066082512272, 2.98054166850019483021196192632, 3.92842817128213055915857811861, 4.73339480470200908725662950753, 6.11675240902167177274167002985, 6.51779602412075469500115003179, 7.35155301191210300128862824800, 8.433991680114459480987162768451, 9.317519993778239477508888710175, 9.719832933752611909378027284309

Graph of the ZZ-function along the critical line