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Properties

Label 2-1110-37.26-c1-0-20
Degree 22
Conductor 11101110
Sign 0.729+0.683i0.729 + 0.683i
Analytic cond. 8.863398.86339
Root an. cond. 2.977142.97714
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  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 3·11-s + (0.499 + 0.866i)12-s + (2.5 − 4.33i)13-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1 − 1.73i)17-s + (0.499 − 0.866i)18-s + (2.5 − 4.33i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 − 0.387i)5-s + 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 0.904·11-s + (0.144 + 0.249i)12-s + (0.693 − 1.20i)13-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.242 − 0.420i)17-s + (0.117 − 0.204i)18-s + (0.573 − 0.993i)19-s + ⋯

Functional equation

Λ(s)=(1110s/2 ΓC(s) L(s)=((0.729+0.683i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1110s/2 ΓC(s+1/2) L(s)=((0.729+0.683i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11101110    =    2⋅3⋅5⋅372 \cdot 3 \cdot 5 \cdot 37
Sign: 0.729+0.683i0.729 + 0.683i
Analytic conductor: 8.863398.86339
Root analytic conductor: 2.977142.97714
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1110(211,⋅)\chi_{1110} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1110, ( :1/2), 0.729+0.683i)(2,\ 1110,\ (\ :1/2),\ 0.729 + 0.683i)

Particular Values

L(1)L(1) ≈\approx 1.9551792151.955179215
L(12)L(\frac12) ≈\approx 1.9551792151.955179215
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
37 1+(−0.5+6.06i)T 1 + (-0.5 + 6.06i)T
good7 1+(−3.5−6.06i)T2 1 + (-3.5 - 6.06i)T^{2}
11 1+3T+11T2 1 + 3T + 11T^{2}
13 1+(−2.5+4.33i)T+(−6.5−11.2i)T2 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.5+4.33i)T+(−9.5−16.4i)T2 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2}
23 1−5T+23T2 1 - 5T + 23T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+6T+31T2 1 + 6T + 31T^{2}
41 1+(1−1.73i)T+(−20.5−35.5i)T2 1 + (1 - 1.73i)T + (-20.5 - 35.5i)T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+3T+47T2 1 + 3T + 47T^{2}
53 1+(1+1.73i)T+(−26.5+45.8i)T2 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2}
59 1+(5.5+9.52i)T+(−29.5+51.0i)T2 1 + (5.5 + 9.52i)T + (-29.5 + 51.0i)T^{2}
61 1+(−3+5.19i)T+(−30.5−52.8i)T2 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1+1.73i)T+(−33.5−58.0i)T2 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2}
71 1+(2−3.46i)T+(−35.5−61.4i)T2 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2}
73 1+4T+73T2 1 + 4T + 73T^{2}
79 1+(−1+1.73i)T+(−39.5−68.4i)T2 1 + (-1 + 1.73i)T + (-39.5 - 68.4i)T^{2}
83 1+(−7−12.1i)T+(−41.5+71.8i)T2 1 + (-7 - 12.1i)T + (-41.5 + 71.8i)T^{2}
89 1+(−2.5−4.33i)T+(−44.5+77.0i)T2 1 + (-2.5 - 4.33i)T + (-44.5 + 77.0i)T^{2}
97 1−16T+97T2 1 - 16T + 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

−9.451527764073938227669520685484, −8.778575515497031681308082004788, −7.952092881369026163692617986427, −7.33284778826714492784211051714, −6.38661684263454426976783781788, −5.43264075326392101073570722249, −4.84736269019168766151916501055, −3.40622043192681713601759837762, −2.54712041799946166243148096999, −0.77068668204527063927287674008, 1.60514424115448976317171268497, 2.76778029490586732208383663897, 3.65289029493915044012674541124, 4.58560061388893452527437944368, 5.52512989908406569372306227121, 6.43070490361064058370796256610, 7.48300381395437310381146618550, 8.575555120512153594599415289318, 9.206007835969312755106643406564, 10.18292189491926584971762581825

Graph of the ZZ-function along the critical line