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Properties

Label 2-1950-13.4-c1-0-7
Degree 22
Conductor 19501950
Sign 0.711−0.702i0.711 - 0.702i
Analytic cond. 15.570815.5708
Root an. cond. 3.945983.94598
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  + (−0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (−0.866 + 0.499i)6-s + (−0.633 + 0.366i)7-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−4.09 − 2.36i)11-s + 0.999·12-s + (−2.59 + 2.5i)13-s + 0.732·14-s + (−0.5 + 0.866i)16-s + (1.13 + 1.96i)17-s + 0.999i·18-s + (−1.09 + 0.633i)19-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.288 − 0.499i)3-s + (0.249 + 0.433i)4-s + (−0.353 + 0.204i)6-s + (−0.239 + 0.138i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (−1.23 − 0.713i)11-s + 0.288·12-s + (−0.720 + 0.693i)13-s + 0.195·14-s + (−0.125 + 0.216i)16-s + (0.275 + 0.476i)17-s + 0.235i·18-s + (−0.251 + 0.145i)19-s + ⋯

Functional equation

Λ(s)=(1950s/2 ΓC(s) L(s)=((0.711−0.702i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1950s/2 ΓC(s+1/2) L(s)=((0.711−0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19501950    =    2⋅3⋅52⋅132 \cdot 3 \cdot 5^{2} \cdot 13
Sign: 0.711−0.702i0.711 - 0.702i
Analytic conductor: 15.570815.5708
Root analytic conductor: 3.945983.94598
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1950(901,⋅)\chi_{1950} (901, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1950, ( :1/2), 0.711−0.702i)(2,\ 1950,\ (\ :1/2),\ 0.711 - 0.702i)

Particular Values

L(1)L(1) ≈\approx 0.79416090290.7941609029
L(12)L(\frac12) ≈\approx 0.79416090290.7941609029
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1 1
13 1+(2.59−2.5i)T 1 + (2.59 - 2.5i)T
good7 1+(0.633−0.366i)T+(3.5−6.06i)T2 1 + (0.633 - 0.366i)T + (3.5 - 6.06i)T^{2}
11 1+(4.09+2.36i)T+(5.5+9.52i)T2 1 + (4.09 + 2.36i)T + (5.5 + 9.52i)T^{2}
17 1+(−1.13−1.96i)T+(−8.5+14.7i)T2 1 + (-1.13 - 1.96i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.09−0.633i)T+(9.5−16.4i)T2 1 + (1.09 - 0.633i)T + (9.5 - 16.4i)T^{2}
23 1+(−3.09+5.36i)T+(−11.5−19.9i)T2 1 + (-3.09 + 5.36i)T + (-11.5 - 19.9i)T^{2}
29 1+(1.23−2.13i)T+(−14.5−25.1i)T2 1 + (1.23 - 2.13i)T + (-14.5 - 25.1i)T^{2}
31 1−5.46iT−31T2 1 - 5.46iT - 31T^{2}
37 1+(−9.06−5.23i)T+(18.5+32.0i)T2 1 + (-9.06 - 5.23i)T + (18.5 + 32.0i)T^{2}
41 1+(−9.86−5.69i)T+(20.5+35.5i)T2 1 + (-9.86 - 5.69i)T + (20.5 + 35.5i)T^{2}
43 1+(3.83+6.63i)T+(−21.5+37.2i)T2 1 + (3.83 + 6.63i)T + (-21.5 + 37.2i)T^{2}
47 1−8.19iT−47T2 1 - 8.19iT - 47T^{2}
53 1+0.464T+53T2 1 + 0.464T + 53T^{2}
59 1+(−6.92+4i)T+(29.5−51.0i)T2 1 + (-6.92 + 4i)T + (29.5 - 51.0i)T^{2}
61 1+(0.598+1.03i)T+(−30.5+52.8i)T2 1 + (0.598 + 1.03i)T + (-30.5 + 52.8i)T^{2}
67 1+(−9.63−5.56i)T+(33.5+58.0i)T2 1 + (-9.63 - 5.56i)T + (33.5 + 58.0i)T^{2}
71 1+(1.09−0.633i)T+(35.5−61.4i)T2 1 + (1.09 - 0.633i)T + (35.5 - 61.4i)T^{2}
73 1−9.73iT−73T2 1 - 9.73iT - 73T^{2}
79 1+9.46T+79T2 1 + 9.46T + 79T^{2}
83 1−10.1iT−83T2 1 - 10.1iT - 83T^{2}
89 1+(−2.19−1.26i)T+(44.5+77.0i)T2 1 + (-2.19 - 1.26i)T + (44.5 + 77.0i)T^{2}
97 1+(5.19−3i)T+(48.5−84.0i)T2 1 + (5.19 - 3i)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

−9.281476123051157153754284441784, −8.364905483913143251447901853029, −7.999473910311689566613891934275, −7.03416989196933358589694681604, −6.34677380575521993574943930300, −5.32376443054217428520377202082, −4.24027703156403770087322553962, −2.95367496971954104060929628859, −2.46784770139233999295301256328, −1.09577675853303949248725458752, 0.37364629654869154881557387663, 2.19564666206182251916709746146, 2.97652263717091875013959214311, 4.24680890881607784523993358335, 5.20613980985200007037442651216, 5.75631243232626555413781224685, 7.05695829845152019930208743135, 7.64643013507142960323129453040, 8.144409329188613570728642150960, 9.361457837363070875867155167040

Graph of the ZZ-function along the critical line