Properties

Label 2-1850-5.4-c1-0-12
Degree 22
Conductor 18501850
Sign 0.4470.894i0.447 - 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 + 0.302i·3-s − 4-s + 0.302·6-s + 4.60i·7-s + i·8-s + 2.90·9-s + 1.30·11-s − 0.302i·12-s + 2.30i·13-s + 4.60·14-s + 16-s − 6i·17-s − 2.90i·18-s − 2·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.174i·3-s − 0.5·4-s + 0.123·6-s + 1.74i·7-s + 0.353i·8-s + 0.969·9-s + 0.392·11-s − 0.0874i·12-s + 0.638i·13-s + 1.23·14-s + 0.250·16-s − 1.45i·17-s − 0.685i·18-s − 0.458·19-s + ⋯

Functional equation

Λ(s)=(1850s/2ΓC(s)L(s)=((0.4470.894i)Λ(2s)\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/2ΓC(s+1/2)L(s)=((0.4470.894i)Λ(1s)\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    =    252372 \cdot 5^{2} \cdot 37
Sign: 0.4470.894i0.447 - 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), 0.4470.894i)(2,\ 1850,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) \approx 1.4717594251.471759425
L(12)L(\frac12) \approx 1.4717594251.471759425
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+iT 1 + iT
5 1 1
37 1iT 1 - iT
good3 10.302iT3T2 1 - 0.302iT - 3T^{2}
7 14.60iT7T2 1 - 4.60iT - 7T^{2}
11 11.30T+11T2 1 - 1.30T + 11T^{2}
13 12.30iT13T2 1 - 2.30iT - 13T^{2}
17 1+6iT17T2 1 + 6iT - 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 16.90iT23T2 1 - 6.90iT - 23T^{2}
29 1+6.90T+29T2 1 + 6.90T + 29T^{2}
31 13.30T+31T2 1 - 3.30T + 31T^{2}
41 1+0.908T+41T2 1 + 0.908T + 41T^{2}
43 16.60iT43T2 1 - 6.60iT - 43T^{2}
47 1+2.60iT47T2 1 + 2.60iT - 47T^{2}
53 16iT53T2 1 - 6iT - 53T^{2}
59 1+3.39T+59T2 1 + 3.39T + 59T^{2}
61 1+10.5T+61T2 1 + 10.5T + 61T^{2}
67 114.5iT67T2 1 - 14.5iT - 67T^{2}
71 16T+71T2 1 - 6T + 71T^{2}
73 18.69iT73T2 1 - 8.69iT - 73T^{2}
79 116.1T+79T2 1 - 16.1T + 79T^{2}
83 1+17.2iT83T2 1 + 17.2iT - 83T^{2}
89 1+5.21T+89T2 1 + 5.21T + 89T^{2}
97 112.4iT97T2 1 - 12.4iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−9.261035256779632435098519895436, −9.107242808474755464919339876143, −7.939258702848124228975144676821, −7.04342717995339147413024972463, −6.02735618369702835618610925151, −5.19516768060739279086933940796, −4.43809123102382197185587370236, −3.37000955572305406815820992691, −2.39197781983994159688185366103, −1.49857958825234531364959375963, 0.56774665187326436386963814195, 1.74670200477888196012967904422, 3.62079778364276613116429032460, 4.12974595855885795438674133190, 4.93171158631216274998438129249, 6.30087272990105668433948560233, 6.64808339352429282709096033937, 7.61022286175580840424156729173, 7.975997597319450431210831673884, 8.986845319187496416613010018571

Graph of the ZZ-function along the critical line