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Properties

Label 2-1740-29.17-c2-0-22
Degree 22
Conductor 17401740
Sign 0.720−0.693i0.720 - 0.693i
Analytic cond. 47.411547.4115
Root an. cond. 6.885606.88560
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 + 1.22i)3-s + 2.23i·5-s + 4.92·7-s − 2.99i·9-s + (−4.50 + 4.50i)11-s + 2.37i·13-s + (−2.73 − 2.73i)15-s + (9.67 − 9.67i)17-s + (3.07 − 3.07i)19-s + (−6.02 + 6.02i)21-s + 20.3·23-s − 5.00·25-s + (3.67 + 3.67i)27-s + (−4.95 − 28.5i)29-s + (31.2 − 31.2i)31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + 0.447i·5-s + 0.702·7-s − 0.333i·9-s + (−0.409 + 0.409i)11-s + 0.183i·13-s + (−0.182 − 0.182i)15-s + (0.569 − 0.569i)17-s + (0.161 − 0.161i)19-s + (−0.286 + 0.286i)21-s + 0.885·23-s − 0.200·25-s + (0.136 + 0.136i)27-s + (−0.170 − 0.985i)29-s + (1.00 − 1.00i)31-s + ⋯

Functional equation

Λ(s)=(1740s/2 ΓC(s) L(s)=((0.720−0.693i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(1740s/2 ΓC(s+1) L(s)=((0.720−0.693i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17401740    =    22⋅3⋅5⋅292^{2} \cdot 3 \cdot 5 \cdot 29
Sign: 0.720−0.693i0.720 - 0.693i
Analytic conductor: 47.411547.4115
Root analytic conductor: 6.885606.88560
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ1740(481,⋅)\chi_{1740} (481, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1740, ( :1), 0.720−0.693i)(2,\ 1740,\ (\ :1),\ 0.720 - 0.693i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 1.8870977341.887097734
L(12)L(\frac12) ≈\approx 1.8870977341.887097734
L(2)L(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 1
3 1+(1.22−1.22i)T 1 + (1.22 - 1.22i)T
5 1−2.23iT 1 - 2.23iT
29 1+(4.95+28.5i)T 1 + (4.95 + 28.5i)T
good7 1−4.92T+49T2 1 - 4.92T + 49T^{2}
11 1+(4.50−4.50i)T−121iT2 1 + (4.50 - 4.50i)T - 121iT^{2}
13 1−2.37iT−169T2 1 - 2.37iT - 169T^{2}
17 1+(−9.67+9.67i)T−289iT2 1 + (-9.67 + 9.67i)T - 289iT^{2}
19 1+(−3.07+3.07i)T−361iT2 1 + (-3.07 + 3.07i)T - 361iT^{2}
23 1−20.3T+529T2 1 - 20.3T + 529T^{2}
31 1+(−31.2+31.2i)T−961iT2 1 + (-31.2 + 31.2i)T - 961iT^{2}
37 1+(−5.24−5.24i)T+1.36e3iT2 1 + (-5.24 - 5.24i)T + 1.36e3iT^{2}
41 1+(−47.2−47.2i)T+1.68e3iT2 1 + (-47.2 - 47.2i)T + 1.68e3iT^{2}
43 1+(51.9−51.9i)T−1.84e3iT2 1 + (51.9 - 51.9i)T - 1.84e3iT^{2}
47 1+(4.56+4.56i)T+2.20e3iT2 1 + (4.56 + 4.56i)T + 2.20e3iT^{2}
53 1−17.4T+2.80e3T2 1 - 17.4T + 2.80e3T^{2}
59 1−60.6T+3.48e3T2 1 - 60.6T + 3.48e3T^{2}
61 1+(−77.4+77.4i)T−3.72e3iT2 1 + (-77.4 + 77.4i)T - 3.72e3iT^{2}
67 1−24.2iT−4.48e3T2 1 - 24.2iT - 4.48e3T^{2}
71 1+63.5iT−5.04e3T2 1 + 63.5iT - 5.04e3T^{2}
73 1+(32.9+32.9i)T+5.32e3iT2 1 + (32.9 + 32.9i)T + 5.32e3iT^{2}
79 1+(−24.8+24.8i)T−6.24e3iT2 1 + (-24.8 + 24.8i)T - 6.24e3iT^{2}
83 1−79.9T+6.88e3T2 1 - 79.9T + 6.88e3T^{2}
89 1+(63.0−63.0i)T−7.92e3iT2 1 + (63.0 - 63.0i)T - 7.92e3iT^{2}
97 1+(47.9+47.9i)T+9.40e3iT2 1 + (47.9 + 47.9i)T + 9.40e3iT^{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

−9.590995726245984946870039526799, −8.277717178201367736287630702839, −7.71602186350652925358705906325, −6.79816511080163856551369280481, −5.97584734537615331559280638660, −5.00106058755262184016067999818, −4.46411479059993638615727040759, −3.26223059846540202122628263232, −2.27193640175100850975630608511, −0.849052439393846156991305048602, 0.75327863446040522808491353251, 1.68102380095239282873275377723, 2.95891538900941435914797183812, 4.10729107553766335825124323428, 5.25526476278162915284799781140, 5.50115528426185268786962762761, 6.71602923441597617264191430564, 7.43884468939588644810343242426, 8.363116187115591691593620802943, 8.714189049447911516546057788216

Graph of the ZZ-function along the critical line