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Properties

Label 2-175-175.103-c3-0-25
Degree 22
Conductor 175175
Sign −0.183−0.982i-0.183 - 0.982i
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.230 + 0.0120i)2-s + (6.35 + 7.84i)3-s + (−7.90 − 0.830i)4-s + (9.14 − 6.42i)5-s + (1.36 + 1.88i)6-s + (14.8 + 11.0i)7-s + (−3.63 − 0.575i)8-s + (−15.5 + 73.3i)9-s + (2.18 − 1.36i)10-s + (−30.1 + 6.40i)11-s + (−43.7 − 67.3i)12-s + (39.6 + 20.2i)13-s + (3.28 + 2.73i)14-s + (108. + 30.9i)15-s + (61.3 + 13.0i)16-s + (−23.6 − 61.4i)17-s + ⋯
L(s)  = 1  + (0.0813 + 0.00426i)2-s + (1.22 + 1.51i)3-s + (−0.987 − 0.103i)4-s + (0.818 − 0.575i)5-s + (0.0930 + 0.128i)6-s + (0.801 + 0.598i)7-s + (−0.160 − 0.0254i)8-s + (−0.577 + 2.71i)9-s + (0.0690 − 0.0433i)10-s + (−0.825 + 0.175i)11-s + (−1.05 − 1.61i)12-s + (0.845 + 0.431i)13-s + (0.0626 + 0.0521i)14-s + (1.86 + 0.532i)15-s + (0.958 + 0.203i)16-s + (−0.336 − 0.877i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.183−0.982i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.183−0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.183−0.982i-0.183 - 0.982i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), −0.183−0.982i)(2,\ 175,\ (\ :3/2),\ -0.183 - 0.982i)

Particular Values

L(2)L(2) ≈\approx 1.56709+1.88755i1.56709 + 1.88755i
L(12)L(\frac12) ≈\approx 1.56709+1.88755i1.56709 + 1.88755i
L(52)L(\frac{5}{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+(−9.14+6.42i)T 1 + (-9.14 + 6.42i)T
7 1+(−14.8−11.0i)T 1 + (-14.8 - 11.0i)T
good2 1+(−0.230−0.0120i)T+(7.95+0.836i)T2 1 + (-0.230 - 0.0120i)T + (7.95 + 0.836i)T^{2}
3 1+(−6.35−7.84i)T+(−5.61+26.4i)T2 1 + (-6.35 - 7.84i)T + (-5.61 + 26.4i)T^{2}
11 1+(30.1−6.40i)T+(1.21e3−541.i)T2 1 + (30.1 - 6.40i)T + (1.21e3 - 541. i)T^{2}
13 1+(−39.6−20.2i)T+(1.29e3+1.77e3i)T2 1 + (-39.6 - 20.2i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(23.6+61.4i)T+(−3.65e3+3.28e3i)T2 1 + (23.6 + 61.4i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−0.806−7.67i)T+(−6.70e3+1.42e3i)T2 1 + (-0.806 - 7.67i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(4.70−89.8i)T+(−1.21e4−1.27e3i)T2 1 + (4.70 - 89.8i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(80.6−110.i)T+(−7.53e3−2.31e4i)T2 1 + (80.6 - 110. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−28.6+64.3i)T+(−1.99e4−2.21e4i)T2 1 + (-28.6 + 64.3i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−9.88+6.42i)T+(2.06e4−4.62e4i)T2 1 + (-9.88 + 6.42i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−60.6+19.6i)T+(5.57e4−4.05e4i)T2 1 + (-60.6 + 19.6i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(179.+179.i)T+7.95e4iT2 1 + (179. + 179. i)T + 7.95e4iT^{2}
47 1+(61.7+23.6i)T+(7.71e4+6.94e4i)T2 1 + (61.7 + 23.6i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−494.+400.i)T+(3.09e4−1.45e5i)T2 1 + (-494. + 400. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(−32.2+35.8i)T+(−2.14e4−2.04e5i)T2 1 + (-32.2 + 35.8i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−539.+485.i)T+(2.37e4−2.25e5i)T2 1 + (-539. + 485. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−870.+334.i)T+(2.23e5−2.01e5i)T2 1 + (-870. + 334. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−663.−481.i)T+(1.10e5+3.40e5i)T2 1 + (-663. - 481. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−573.+882.i)T+(−1.58e5−3.55e5i)T2 1 + (-573. + 882. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−60.3−135.i)T+(−3.29e5+3.66e5i)T2 1 + (-60.3 - 135. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−74.9+473.i)T+(−5.43e5−1.76e5i)T2 1 + (-74.9 + 473. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(712.+791.i)T+(−7.36e4+7.01e5i)T2 1 + (712. + 791. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(40.3+254.i)T+(−8.68e5+2.82e5i)T2 1 + (40.3 + 254. i)T + (-8.68e5 + 2.82e5i)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

−13.00474300185756091876008448575, −11.24679335044902342613729932270, −10.12180191430162074077723839738, −9.375861895655212551460249562984, −8.754049725336695059477943471054, −8.048527640192367537859413955712, −5.34771706550116451563354158001, −4.95716168550239782445649668723, −3.72503430188512512783422327999, −2.15635115284315395791236502234, 1.06368100613493993662943829206, 2.46684127015152994724996782479, 3.80431498962464488614268317916, 5.71854915594240041138618825599, 6.90576932518495189248406243550, 8.122708402443772609680378808972, 8.472941007584600804862781709604, 9.764242945550491825945769084294, 10.94876361095732128275929655220, 12.52243540707160588388467882004

Graph of the ZZ-function along the critical line