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Properties

Label 2-665-19.7-c1-0-25
Degree 22
Conductor 665665
Sign 0.999−0.0375i0.999 - 0.0375i
Analytic cond. 5.310055.31005
Root an. cond. 2.304352.30435
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  + (0.302 + 0.523i)2-s + (0.354 + 0.614i)3-s + (0.817 − 1.41i)4-s + (0.5 + 0.866i)5-s + (−0.214 + 0.371i)6-s − 7-s + 2.19·8-s + (1.24 − 2.16i)9-s + (−0.302 + 0.523i)10-s − 0.340·11-s + 1.15·12-s + (3.17 − 5.49i)13-s + (−0.302 − 0.523i)14-s + (−0.354 + 0.614i)15-s + (−0.969 − 1.67i)16-s + (2.54 + 4.41i)17-s + ⋯
L(s)  = 1  + (0.213 + 0.370i)2-s + (0.204 + 0.354i)3-s + (0.408 − 0.707i)4-s + (0.223 + 0.387i)5-s + (−0.0875 + 0.151i)6-s − 0.377·7-s + 0.777·8-s + (0.416 − 0.720i)9-s + (−0.0956 + 0.165i)10-s − 0.102·11-s + 0.334·12-s + (0.879 − 1.52i)13-s + (−0.0808 − 0.139i)14-s + (−0.0915 + 0.158i)15-s + (−0.242 − 0.419i)16-s + (0.618 + 1.07i)17-s + ⋯

Functional equation

Λ(s)=(665s/2 ΓC(s) L(s)=((0.999−0.0375i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(665s/2 ΓC(s+1/2) L(s)=((0.999−0.0375i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 665 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0375i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 665665    =    5⋅7⋅195 \cdot 7 \cdot 19
Sign: 0.999−0.0375i0.999 - 0.0375i
Analytic conductor: 5.310055.31005
Root analytic conductor: 2.304352.30435
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ665(596,⋅)\chi_{665} (596, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 665, ( :1/2), 0.999−0.0375i)(2,\ 665,\ (\ :1/2),\ 0.999 - 0.0375i)

Particular Values

L(1)L(1) ≈\approx 2.14762+0.0403443i2.14762 + 0.0403443i
L(12)L(\frac12) ≈\approx 2.14762+0.0403443i2.14762 + 0.0403443i
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)
bad5 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1+T 1 + T
19 1+(4.22−1.08i)T 1 + (4.22 - 1.08i)T
good2 1+(−0.302−0.523i)T+(−1+1.73i)T2 1 + (-0.302 - 0.523i)T + (-1 + 1.73i)T^{2}
3 1+(−0.354−0.614i)T+(−1.5+2.59i)T2 1 + (-0.354 - 0.614i)T + (-1.5 + 2.59i)T^{2}
11 1+0.340T+11T2 1 + 0.340T + 11T^{2}
13 1+(−3.17+5.49i)T+(−6.5−11.2i)T2 1 + (-3.17 + 5.49i)T + (-6.5 - 11.2i)T^{2}
17 1+(−2.54−4.41i)T+(−8.5+14.7i)T2 1 + (-2.54 - 4.41i)T + (-8.5 + 14.7i)T^{2}
23 1+(−2.28+3.95i)T+(−11.5−19.9i)T2 1 + (-2.28 + 3.95i)T + (-11.5 - 19.9i)T^{2}
29 1+(4.46−7.73i)T+(−14.5−25.1i)T2 1 + (4.46 - 7.73i)T + (-14.5 - 25.1i)T^{2}
31 1−6.98T+31T2 1 - 6.98T + 31T^{2}
37 1+5.09T+37T2 1 + 5.09T + 37T^{2}
41 1+(−5.15−8.92i)T+(−20.5+35.5i)T2 1 + (-5.15 - 8.92i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.54+2.67i)T+(−21.5+37.2i)T2 1 + (1.54 + 2.67i)T + (-21.5 + 37.2i)T^{2}
47 1+(−3.43+5.94i)T+(−23.5−40.7i)T2 1 + (-3.43 + 5.94i)T + (-23.5 - 40.7i)T^{2}
53 1+(3.62−6.28i)T+(−26.5−45.8i)T2 1 + (3.62 - 6.28i)T + (-26.5 - 45.8i)T^{2}
59 1+(6.28+10.8i)T+(−29.5+51.0i)T2 1 + (6.28 + 10.8i)T + (-29.5 + 51.0i)T^{2}
61 1+(4.62−8.00i)T+(−30.5−52.8i)T2 1 + (4.62 - 8.00i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.37+2.38i)T+(−33.5−58.0i)T2 1 + (-1.37 + 2.38i)T + (-33.5 - 58.0i)T^{2}
71 1+(−1.65−2.86i)T+(−35.5+61.4i)T2 1 + (-1.65 - 2.86i)T + (-35.5 + 61.4i)T^{2}
73 1+(−4.01−6.94i)T+(−36.5+63.2i)T2 1 + (-4.01 - 6.94i)T + (-36.5 + 63.2i)T^{2}
79 1+(1.89+3.28i)T+(−39.5+68.4i)T2 1 + (1.89 + 3.28i)T + (-39.5 + 68.4i)T^{2}
83 1−2.93T+83T2 1 - 2.93T + 83T^{2}
89 1+(8.59−14.8i)T+(−44.5−77.0i)T2 1 + (8.59 - 14.8i)T + (-44.5 - 77.0i)T^{2}
97 1+(−1.81−3.14i)T+(−48.5+84.0i)T2 1 + (-1.81 - 3.14i)T + (-48.5 + 84.0i)T^{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

−10.50600121850357771497726585850, −9.922391225117823905525177631498, −8.797610185222131381213749753046, −7.86270833804282339900361019643, −6.65351560420198975556681616470, −6.17603827827216047495803920902, −5.24652873657744941649466390605, −3.92929924419163782390043988092, −2.92389515730358385652827210690, −1.25235284977949843713114010027, 1.63181214127192164033378758206, 2.60899021746538729959700511927, 3.90572955083752318576605638144, 4.75482397172763060754111902012, 6.18239824084027760009457030798, 7.09344646910155609624498347766, 7.80816211374419249522723350421, 8.780856771309166145527580194621, 9.599423428870587885710224796719, 10.71254161288893698242392280649

Graph of the ZZ-function along the critical line