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Properties

Label 2-150-15.8-c1-0-4
Degree 22
Conductor 150150
Sign 0.828−0.559i0.828 - 0.559i
Analytic cond. 1.197751.19775
Root an. cond. 1.094421.09442
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 + (1.67 − 0.448i)3-s + 1.00i·4-s + (1.5 + 0.866i)6-s + (−2.44 + 2.44i)7-s + (−0.707 + 0.707i)8-s + (2.59 − 1.50i)9-s − 5.19i·11-s + (0.448 + 1.67i)12-s − 3.46·14-s − 1.00·16-s + (−2.12 − 2.12i)17-s + (2.89 + 0.776i)18-s + i·19-s + (−3 + 5.19i)21-s + (3.67 − 3.67i)22-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.965 − 0.258i)3-s + 0.500i·4-s + (0.612 + 0.353i)6-s + (−0.925 + 0.925i)7-s + (−0.250 + 0.250i)8-s + (0.866 − 0.5i)9-s − 1.56i·11-s + (0.129 + 0.482i)12-s − 0.925·14-s − 0.250·16-s + (−0.514 − 0.514i)17-s + (0.683 + 0.183i)18-s + 0.229i·19-s + (−0.654 + 1.13i)21-s + (0.783 − 0.783i)22-s + ⋯

Functional equation

Λ(s)=(150s/2 ΓC(s) L(s)=((0.828−0.559i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(150s/2 ΓC(s+1/2) L(s)=((0.828−0.559i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 150150    =    2⋅3⋅522 \cdot 3 \cdot 5^{2}
Sign: 0.828−0.559i0.828 - 0.559i
Analytic conductor: 1.197751.19775
Root analytic conductor: 1.094421.09442
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ150(143,⋅)\chi_{150} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 150, ( :1/2), 0.828−0.559i)(2,\ 150,\ (\ :1/2),\ 0.828 - 0.559i)

Particular Values

L(1)L(1) ≈\approx 1.63019+0.499193i1.63019 + 0.499193i
L(12)L(\frac12) ≈\approx 1.63019+0.499193i1.63019 + 0.499193i
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
3 1+(−1.67+0.448i)T 1 + (-1.67 + 0.448i)T
5 1 1
good7 1+(2.44−2.44i)T−7iT2 1 + (2.44 - 2.44i)T - 7iT^{2}
11 1+5.19iT−11T2 1 + 5.19iT - 11T^{2}
13 1+13iT2 1 + 13iT^{2}
17 1+(2.12+2.12i)T+17iT2 1 + (2.12 + 2.12i)T + 17iT^{2}
19 1−iT−19T2 1 - iT - 19T^{2}
23 1+(4.24−4.24i)T−23iT2 1 + (4.24 - 4.24i)T - 23iT^{2}
29 1+29T2 1 + 29T^{2}
31 1+2T+31T2 1 + 2T + 31T^{2}
37 1+(−2.44+2.44i)T−37iT2 1 + (-2.44 + 2.44i)T - 37iT^{2}
41 1−5.19iT−41T2 1 - 5.19iT - 41T^{2}
43 1+(−2.44−2.44i)T+43iT2 1 + (-2.44 - 2.44i)T + 43iT^{2}
47 1+47iT2 1 + 47iT^{2}
53 1+(−4.24+4.24i)T−53iT2 1 + (-4.24 + 4.24i)T - 53iT^{2}
59 1+10.3T+59T2 1 + 10.3T + 59T^{2}
61 1−14T+61T2 1 - 14T + 61T^{2}
67 1+(3.67−3.67i)T−67iT2 1 + (3.67 - 3.67i)T - 67iT^{2}
71 1−71T2 1 - 71T^{2}
73 1+(−6.12−6.12i)T+73iT2 1 + (-6.12 - 6.12i)T + 73iT^{2}
79 1−14iT−79T2 1 - 14iT - 79T^{2}
83 1+(−2.12+2.12i)T−83iT2 1 + (-2.12 + 2.12i)T - 83iT^{2}
89 1−15.5T+89T2 1 - 15.5T + 89T^{2}
97 1+(4.89−4.89i)T−97iT2 1 + (4.89 - 4.89i)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

−13.33061749048803286923780687149, −12.45726626675015378193100454794, −11.36671939821205130573798369171, −9.668539346855621596939464055384, −8.833714664259009541260787522317, −7.927639899796082279395426842762, −6.59019295305139983385951764781, −5.65662929676451807619200184444, −3.75160830104294674073228019702, −2.70572854985074972695335552668, 2.21477996751888029364567173423, 3.74289328770649006938144316249, 4.59037963425345594477193943323, 6.57907450940759323929561227870, 7.57105743245793777515742952631, 9.082740597853566796540835246490, 10.02898589089507563784961585740, 10.58104763866450892990411159604, 12.27648274533574283403614565765, 13.00216077312138081701779265833

Graph of the ZZ-function along the critical line