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Properties

Label 2-175-175.103-c3-0-13
Degree 22
Conductor 175175
Sign −0.938−0.346i-0.938 - 0.346i
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  + (1.48 + 0.0777i)2-s + (2.51 + 3.10i)3-s + (−5.76 − 0.605i)4-s + (4.25 + 10.3i)5-s + (3.48 + 4.79i)6-s + (−12.4 + 13.7i)7-s + (−20.2 − 3.20i)8-s + (2.30 − 10.8i)9-s + (5.50 + 15.6i)10-s + (−33.2 + 7.06i)11-s + (−12.5 − 19.3i)12-s + (−13.9 − 7.08i)13-s + (−19.4 + 19.4i)14-s + (−21.3 + 39.1i)15-s + (15.5 + 3.30i)16-s + (−27.0 − 70.3i)17-s + ⋯
L(s)  = 1  + (0.524 + 0.0274i)2-s + (0.483 + 0.596i)3-s + (−0.720 − 0.0756i)4-s + (0.380 + 0.924i)5-s + (0.237 + 0.326i)6-s + (−0.670 + 0.741i)7-s + (−0.894 − 0.141i)8-s + (0.0853 − 0.401i)9-s + (0.174 + 0.495i)10-s + (−0.911 + 0.193i)11-s + (−0.302 − 0.466i)12-s + (−0.296 − 0.151i)13-s + (−0.372 + 0.370i)14-s + (−0.368 + 0.673i)15-s + (0.242 + 0.0516i)16-s + (−0.385 − 1.00i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.938−0.346i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.938−0.346i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.938−0.346i-0.938 - 0.346i
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), −0.938−0.346i)(2,\ 175,\ (\ :3/2),\ -0.938 - 0.346i)

Particular Values

L(2)L(2) ≈\approx 0.210757+1.17916i0.210757 + 1.17916i
L(12)L(\frac12) ≈\approx 0.210757+1.17916i0.210757 + 1.17916i
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+(−4.25−10.3i)T 1 + (-4.25 - 10.3i)T
7 1+(12.4−13.7i)T 1 + (12.4 - 13.7i)T
good2 1+(−1.48−0.0777i)T+(7.95+0.836i)T2 1 + (-1.48 - 0.0777i)T + (7.95 + 0.836i)T^{2}
3 1+(−2.51−3.10i)T+(−5.61+26.4i)T2 1 + (-2.51 - 3.10i)T + (-5.61 + 26.4i)T^{2}
11 1+(33.2−7.06i)T+(1.21e3−541.i)T2 1 + (33.2 - 7.06i)T + (1.21e3 - 541. i)T^{2}
13 1+(13.9+7.08i)T+(1.29e3+1.77e3i)T2 1 + (13.9 + 7.08i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(27.0+70.3i)T+(−3.65e3+3.28e3i)T2 1 + (27.0 + 70.3i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−3.94−37.5i)T+(−6.70e3+1.42e3i)T2 1 + (-3.94 - 37.5i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(10.3−197.i)T+(−1.21e4−1.27e3i)T2 1 + (10.3 - 197. i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(33.8−46.6i)T+(−7.53e3−2.31e4i)T2 1 + (33.8 - 46.6i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(69.5−156.i)T+(−1.99e4−2.21e4i)T2 1 + (69.5 - 156. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−48.0+31.2i)T+(2.06e4−4.62e4i)T2 1 + (-48.0 + 31.2i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(17.8−5.78i)T+(5.57e4−4.05e4i)T2 1 + (17.8 - 5.78i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−1.93−1.93i)T+7.95e4iT2 1 + (-1.93 - 1.93i)T + 7.95e4iT^{2}
47 1+(−535.−205.i)T+(7.71e4+6.94e4i)T2 1 + (-535. - 205. i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(334.−271.i)T+(3.09e4−1.45e5i)T2 1 + (334. - 271. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(215.−239.i)T+(−2.14e4−2.04e5i)T2 1 + (215. - 239. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(355.−320.i)T+(2.37e4−2.25e5i)T2 1 + (355. - 320. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−720.+276.i)T+(2.23e5−2.01e5i)T2 1 + (-720. + 276. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−617.−448.i)T+(1.10e5+3.40e5i)T2 1 + (-617. - 448. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−20.1+31.0i)T+(−1.58e5−3.55e5i)T2 1 + (-20.1 + 31.0i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(294.+662.i)T+(−3.29e5+3.66e5i)T2 1 + (294. + 662. i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(−227.+1.43e3i)T+(−5.43e5−1.76e5i)T2 1 + (-227. + 1.43e3i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(−734.−815.i)T+(−7.36e4+7.01e5i)T2 1 + (-734. - 815. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(96.4+609.i)T+(−8.68e5+2.82e5i)T2 1 + (96.4 + 609. 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

−12.86607757279030648157196596565, −11.90497349934201635051983916792, −10.44593757171223682120939741160, −9.545685993872558889789129980679, −9.081886042521281844962145011983, −7.46422780552423942415803353876, −6.07890690616351800224560330341, −5.12230524558620231194843678044, −3.58297326273178639546240882712, −2.77000840651938107780579127008, 0.42174017673805608260217994866, 2.41383458604246806086513556907, 4.07896833964963224778327328824, 5.08549892155989603016269598277, 6.38286201754586518319478229837, 7.86438447423932726051754584456, 8.619892479278414319372452562387, 9.693995008624657875305805904550, 10.73149232516426900647403863546, 12.58069227741889351474484443867

Graph of the ZZ-function along the critical line