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Properties

Label 2-175-175.109-c1-0-0
Degree 22
Conductor 175175
Sign −0.987+0.157i-0.987 + 0.157i
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
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  + (−1.93 − 0.203i)2-s + (−0.500 + 2.35i)3-s + (1.75 + 0.373i)4-s + (−0.306 + 2.21i)5-s + (1.45 − 4.46i)6-s + (−2.52 + 0.802i)7-s + (0.377 + 0.122i)8-s + (−2.56 − 1.14i)9-s + (1.04 − 4.22i)10-s + (2.26 − 1.00i)11-s + (−1.76 + 3.95i)12-s + (−2.29 − 3.16i)13-s + (5.04 − 1.04i)14-s + (−5.06 − 1.83i)15-s + (−3.98 − 1.77i)16-s + (−2.30 + 2.07i)17-s + ⋯
L(s)  = 1  + (−1.37 − 0.144i)2-s + (−0.289 + 1.36i)3-s + (0.878 + 0.186i)4-s + (−0.137 + 0.990i)5-s + (0.592 − 1.82i)6-s + (−0.952 + 0.303i)7-s + (0.133 + 0.0433i)8-s + (−0.854 − 0.380i)9-s + (0.330 − 1.33i)10-s + (0.683 − 0.304i)11-s + (−0.508 + 1.14i)12-s + (−0.637 − 0.876i)13-s + (1.34 − 0.278i)14-s + (−1.30 − 0.472i)15-s + (−0.997 − 0.443i)16-s + (−0.558 + 0.502i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.987+0.157i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.987+0.157i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.987+0.157i-0.987 + 0.157i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(109,⋅)\chi_{175} (109, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.987+0.157i)(2,\ 175,\ (\ :1/2),\ -0.987 + 0.157i)

Particular Values

L(1)L(1) ≈\approx 0.0220152−0.277325i0.0220152 - 0.277325i
L(12)L(\frac12) ≈\approx 0.0220152−0.277325i0.0220152 - 0.277325i
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+(0.306−2.21i)T 1 + (0.306 - 2.21i)T
7 1+(2.52−0.802i)T 1 + (2.52 - 0.802i)T
good2 1+(1.93+0.203i)T+(1.95+0.415i)T2 1 + (1.93 + 0.203i)T + (1.95 + 0.415i)T^{2}
3 1+(0.500−2.35i)T+(−2.74−1.22i)T2 1 + (0.500 - 2.35i)T + (-2.74 - 1.22i)T^{2}
11 1+(−2.26+1.00i)T+(7.36−8.17i)T2 1 + (-2.26 + 1.00i)T + (7.36 - 8.17i)T^{2}
13 1+(2.29+3.16i)T+(−4.01+12.3i)T2 1 + (2.29 + 3.16i)T + (-4.01 + 12.3i)T^{2}
17 1+(2.30−2.07i)T+(1.77−16.9i)T2 1 + (2.30 - 2.07i)T + (1.77 - 16.9i)T^{2}
19 1+(5.00−1.06i)T+(17.3−7.72i)T2 1 + (5.00 - 1.06i)T + (17.3 - 7.72i)T^{2}
23 1+(−4.53−0.476i)T+(22.4+4.78i)T2 1 + (-4.53 - 0.476i)T + (22.4 + 4.78i)T^{2}
29 1+(0.797+2.45i)T+(−23.4+17.0i)T2 1 + (0.797 + 2.45i)T + (-23.4 + 17.0i)T^{2}
31 1+(−4.22−4.68i)T+(−3.24+30.8i)T2 1 + (-4.22 - 4.68i)T + (-3.24 + 30.8i)T^{2}
37 1+(−3.31+7.45i)T+(−24.7−27.4i)T2 1 + (-3.31 + 7.45i)T + (-24.7 - 27.4i)T^{2}
41 1+(4.21−3.06i)T+(12.6−38.9i)T2 1 + (4.21 - 3.06i)T + (12.6 - 38.9i)T^{2}
43 1−5.80iT−43T2 1 - 5.80iT - 43T^{2}
47 1+(1.61+1.45i)T+(4.91+46.7i)T2 1 + (1.61 + 1.45i)T + (4.91 + 46.7i)T^{2}
53 1+(1.59−7.51i)T+(−48.4−21.5i)T2 1 + (1.59 - 7.51i)T + (-48.4 - 21.5i)T^{2}
59 1+(−1.15−10.9i)T+(−57.7+12.2i)T2 1 + (-1.15 - 10.9i)T + (-57.7 + 12.2i)T^{2}
61 1+(0.904−8.60i)T+(−59.6−12.6i)T2 1 + (0.904 - 8.60i)T + (-59.6 - 12.6i)T^{2}
67 1+(6.37−5.73i)T+(7.00−66.6i)T2 1 + (6.37 - 5.73i)T + (7.00 - 66.6i)T^{2}
71 1+(−4.36−13.4i)T+(−57.4+41.7i)T2 1 + (-4.36 - 13.4i)T + (-57.4 + 41.7i)T^{2}
73 1+(5.00+11.2i)T+(−48.8+54.2i)T2 1 + (5.00 + 11.2i)T + (-48.8 + 54.2i)T^{2}
79 1+(4.18−4.64i)T+(−8.25−78.5i)T2 1 + (4.18 - 4.64i)T + (-8.25 - 78.5i)T^{2}
83 1+(4.78+1.55i)T+(67.1+48.7i)T2 1 + (4.78 + 1.55i)T + (67.1 + 48.7i)T^{2}
89 1+(−1.45+13.8i)T+(−87.0−18.5i)T2 1 + (-1.45 + 13.8i)T + (-87.0 - 18.5i)T^{2}
97 1+(−4.30+1.39i)T+(78.4−57.0i)T2 1 + (-4.30 + 1.39i)T + (78.4 - 57.0i)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

−13.12032663179534705741128062024, −11.66440954314056532569330769147, −10.66251076640813573904393715394, −10.28778477419286522511078935707, −9.434152473644546413389739906349, −8.584596368519048995415364905433, −7.16393189746003631845950389847, −6.00563367557171652209663636521, −4.27315562640448709378033269198, −2.86434998435255576309442903969, 0.40408292363842818713151899360, 1.86447933462948192018786262382, 4.51365107214742066804641230000, 6.57067602543534170820778710031, 6.95862914668735823054224758928, 8.130507436446994962723549901074, 9.099976160287067659101188505294, 9.778669961186532460850247794936, 11.26608343130505456841609132538, 12.21614996041467291788905158993

Graph of the ZZ-function along the critical line