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Properties

Label 2-950-19.7-c1-0-2
Degree 22
Conductor 950950
Sign 0.321−0.946i0.321 - 0.946i
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s − 2·7-s + 0.999·8-s + (1 − 1.73i)9-s + 0.999·12-s + (−3 + 5.19i)13-s + (1 + 1.73i)14-s + (−0.5 − 0.866i)16-s + (−3.5 − 6.06i)17-s − 2·18-s + (3.5 + 2.59i)19-s + (1 + 1.73i)21-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s − 0.755·7-s + 0.353·8-s + (0.333 − 0.577i)9-s + 0.288·12-s + (−0.832 + 1.44i)13-s + (0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s + (−0.848 − 1.47i)17-s − 0.471·18-s + (0.802 + 0.596i)19-s + (0.218 + 0.377i)21-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((0.321−0.946i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((0.321−0.946i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 0.321−0.946i0.321 - 0.946i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(501,⋅)\chi_{950} (501, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 0.321−0.946i)(2,\ 950,\ (\ :1/2),\ 0.321 - 0.946i)

Particular Values

L(1)L(1) ≈\approx 0.289940+0.207644i0.289940 + 0.207644i
L(12)L(\frac12) ≈\approx 0.289940+0.207644i0.289940 + 0.207644i
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+(−3.5−2.59i)T 1 + (-3.5 - 2.59i)T
good3 1+(0.5+0.866i)T+(−1.5+2.59i)T2 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2}
7 1+2T+7T2 1 + 2T + 7T^{2}
11 1+11T2 1 + 11T^{2}
13 1+(3−5.19i)T+(−6.5−11.2i)T2 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2}
17 1+(3.5+6.06i)T+(−8.5+14.7i)T2 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2}
23 1+(1−1.73i)T+(−11.5−19.9i)T2 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2}
29 1+(5−8.66i)T+(−14.5−25.1i)T2 1 + (5 - 8.66i)T + (-14.5 - 25.1i)T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1−4T+37T2 1 - 4T + 37T^{2}
41 1+(1+1.73i)T+(−20.5+35.5i)T2 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6−10.3i)T+(−21.5+37.2i)T2 1 + (-6 - 10.3i)T + (-21.5 + 37.2i)T^{2}
47 1+(−23.5−40.7i)T2 1 + (-23.5 - 40.7i)T^{2}
53 1+(−26.5−45.8i)T2 1 + (-26.5 - 45.8i)T^{2}
59 1+(−0.5−0.866i)T+(−29.5+51.0i)T2 1 + (-0.5 - 0.866i)T + (-29.5 + 51.0i)T^{2}
61 1+(4−6.92i)T+(−30.5−52.8i)T2 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1+(−6−10.3i)T+(−35.5+61.4i)T2 1 + (-6 - 10.3i)T + (-35.5 + 61.4i)T^{2}
73 1+(−1.5−2.59i)T+(−36.5+63.2i)T2 1 + (-1.5 - 2.59i)T + (-36.5 + 63.2i)T^{2}
79 1+(−2−3.46i)T+(−39.5+68.4i)T2 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2}
83 1+13T+83T2 1 + 13T + 83T^{2}
89 1+(−6.5+11.2i)T+(−44.5−77.0i)T2 1 + (-6.5 + 11.2i)T + (-44.5 - 77.0i)T^{2}
97 1+(7.5+12.9i)T+(−48.5+84.0i)T2 1 + (7.5 + 12.9i)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

−9.945375299050809169687558083325, −9.458727408915369424747099895819, −8.929679286484857229234258393949, −7.30412621269998237350727960862, −7.15488639386538675401656096160, −6.05960659677443724933872505973, −4.80830198253135110570329868018, −3.76820332631954001470310510809, −2.64898730943208923543592352889, −1.38495792163468881428220514014, 0.19823096282507986237701611881, 2.25674862786290202247811211318, 3.69883378923026996499869594817, 4.74498306550593141600686354744, 5.62477594214103091033009849738, 6.36835858304964599464030295006, 7.50708464308209509550715752550, 8.012359466433770125134083866667, 9.183689866291951142782107207013, 9.863539739500078601203581498044

Graph of the ZZ-function along the critical line