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Properties

Label 2-950-95.18-c1-0-18
Degree 22
Conductor 950950
Sign 0.446−0.894i0.446 - 0.894i
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (1 − i)7-s + (−0.707 + 0.707i)8-s + 3i·9-s + 2·11-s + (4.24 − 4.24i)13-s + 1.41·14-s − 1.00·16-s + (−3 + 3i)17-s + (−2.12 + 2.12i)18-s + (4.24 + i)19-s + (1.41 + 1.41i)22-s + (−1 − i)23-s + 6·26-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.377 − 0.377i)7-s + (−0.250 + 0.250i)8-s + i·9-s + 0.603·11-s + (1.17 − 1.17i)13-s + 0.377·14-s − 0.250·16-s + (−0.727 + 0.727i)17-s + (−0.499 + 0.499i)18-s + (0.973 + 0.229i)19-s + (0.301 + 0.301i)22-s + (−0.208 − 0.208i)23-s + 1.17·26-s + ⋯

Functional equation

Λ(s)=(950s/2 ΓC(s) L(s)=((0.446−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(950s/2 ΓC(s+1/2) L(s)=((0.446−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 950950    =    2⋅52⋅192 \cdot 5^{2} \cdot 19
Sign: 0.446−0.894i0.446 - 0.894i
Analytic conductor: 7.585787.58578
Root analytic conductor: 2.754232.75423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ950(493,⋅)\chi_{950} (493, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 950, ( :1/2), 0.446−0.894i)(2,\ 950,\ (\ :1/2),\ 0.446 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.93833+1.19841i1.93833 + 1.19841i
L(12)L(\frac12) ≈\approx 1.93833+1.19841i1.93833 + 1.19841i
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1 1
19 1+(−4.24−i)T 1 + (-4.24 - i)T
good3 1−3iT2 1 - 3iT^{2}
7 1+(−1+i)T−7iT2 1 + (-1 + i)T - 7iT^{2}
11 1−2T+11T2 1 - 2T + 11T^{2}
13 1+(−4.24+4.24i)T−13iT2 1 + (-4.24 + 4.24i)T - 13iT^{2}
17 1+(3−3i)T−17iT2 1 + (3 - 3i)T - 17iT^{2}
23 1+(1+i)T+23iT2 1 + (1 + i)T + 23iT^{2}
29 1+29T2 1 + 29T^{2}
31 1−8.48iT−31T2 1 - 8.48iT - 31T^{2}
37 1+(−4.24−4.24i)T+37iT2 1 + (-4.24 - 4.24i)T + 37iT^{2}
41 1+8.48iT−41T2 1 + 8.48iT - 41T^{2}
43 1+(−5−5i)T+43iT2 1 + (-5 - 5i)T + 43iT^{2}
47 1+(7−7i)T−47iT2 1 + (7 - 7i)T - 47iT^{2}
53 1+(−4.24+4.24i)T−53iT2 1 + (-4.24 + 4.24i)T - 53iT^{2}
59 1+59T2 1 + 59T^{2}
61 1−8T+61T2 1 - 8T + 61T^{2}
67 1+(8.48+8.48i)T+67iT2 1 + (8.48 + 8.48i)T + 67iT^{2}
71 1+8.48iT−71T2 1 + 8.48iT - 71T^{2}
73 1+(−1−i)T+73iT2 1 + (-1 - i)T + 73iT^{2}
79 1−8.48T+79T2 1 - 8.48T + 79T^{2}
83 1+(9+9i)T+83iT2 1 + (9 + 9i)T + 83iT^{2}
89 1−8.48T+89T2 1 - 8.48T + 89T^{2}
97 1+(4.24+4.24i)T+97iT2 1 + (4.24 + 4.24i)T + 97iT^{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

−10.47453495126117408397793862778, −9.161951939184174383407841428721, −8.217095982245246452738779843562, −7.78619783504216740247987744881, −6.69472329762435480660431454351, −5.84566808015665426875870241924, −4.97875338710202792318699307782, −4.04032593016334732883283840782, −3.02850197860161588080392427030, −1.44807817494085330301590508190, 1.09224996598398599062437472727, 2.37226409512794249815340943665, 3.68913073477991881696100971166, 4.30845811282412531923279970750, 5.53988068858311242324984151069, 6.36513902269706337491651665572, 7.10996802837178866673510985507, 8.480261320899370882769845778311, 9.267625998531578727949606906722, 9.694250279172984659403977935371

Graph of the ZZ-function along the critical line