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Properties

Label 2-1850-5.4-c1-0-30
Degree 22
Conductor 18501850
Sign −0.447+0.894i-0.447 + 0.894i
Analytic cond. 14.772314.7723
Root an. cond. 3.843473.84347
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  − i·2-s − 2i·3-s − 4-s − 2·6-s + i·7-s + i·8-s − 9-s + 3·11-s + 2i·12-s + 14-s + 16-s + 3i·17-s + i·18-s + 6·19-s + 2·21-s − 3i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s + 0.235i·18-s + 1.37·19-s + 0.436·21-s − 0.639i·22-s + ⋯

Functional equation

Λ(s)=(1850s/2 ΓC(s) L(s)=((−0.447+0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1850s/2 ΓC(s+1/2) L(s)=((−0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18501850    =    2⋅52⋅372 \cdot 5^{2} \cdot 37
Sign: −0.447+0.894i-0.447 + 0.894i
Analytic conductor: 14.772314.7723
Root analytic conductor: 3.843473.84347
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1850(149,⋅)\chi_{1850} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1850, ( :1/2), −0.447+0.894i)(2,\ 1850,\ (\ :1/2),\ -0.447 + 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.8471741871.847174187
L(12)L(\frac12) ≈\approx 1.8471741871.847174187
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+iT 1 + iT
5 1 1
37 1+iT 1 + iT
good3 1+2iT−3T2 1 + 2iT - 3T^{2}
7 1−iT−7T2 1 - iT - 7T^{2}
11 1−3T+11T2 1 - 3T + 11T^{2}
13 1−13T2 1 - 13T^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1−6T+19T2 1 - 6T + 19T^{2}
23 1+2iT−23T2 1 + 2iT - 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
31 1−3T+31T2 1 - 3T + 31T^{2}
41 1−3T+41T2 1 - 3T + 41T^{2}
43 1−iT−43T2 1 - iT - 43T^{2}
47 1−4iT−47T2 1 - 4iT - 47T^{2}
53 1+13iT−53T2 1 + 13iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+15T+61T2 1 + 15T + 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1+2T+71T2 1 + 2T + 71T^{2}
73 1−73T2 1 - 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1−4iT−83T2 1 - 4iT - 83T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1+7iT−97T2 1 + 7iT - 97T^{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

−9.016878666747083129441069379438, −8.199623803560296612358939184017, −7.49469940747098485113384477479, −6.58290769860461660036032458039, −5.94186547462529206669173945191, −4.82798522752783592425277029904, −3.80332166023422941910161875015, −2.76612750318482204677382224436, −1.74917883815201447932811504155, −0.901249338822713350354124349918, 1.10782098922167989188044175594, 3.03803063902427134660923114549, 3.90074513308399965392210705678, 4.62444299874089903314541316567, 5.33841908241949538187458525476, 6.28700624941458639152996448709, 7.19778619434054481492140435772, 7.79810753547288837223010982022, 9.075300239469897935271370439783, 9.269990576951351258150809431634

Graph of the ZZ-function along the critical line