↑

Properties

Label 2-175-175.103-c3-0-24
Degree 22
Conductor 175175
Sign −0.185+0.982i-0.185 + 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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.65 − 0.191i)2-s + (−3.56 − 4.40i)3-s + (5.37 + 0.565i)4-s + (−7.45 − 8.33i)5-s + (12.2 + 16.7i)6-s + (5.89 + 17.5i)7-s + (9.37 + 1.48i)8-s + (−1.07 + 5.04i)9-s + (25.6 + 31.9i)10-s + (45.0 − 9.57i)11-s + (−16.7 − 25.7i)12-s + (34.9 + 17.8i)13-s + (−18.1 − 65.3i)14-s + (−10.1 + 62.5i)15-s + (−76.3 − 16.2i)16-s + (19.9 + 51.9i)17-s + ⋯
L(s)  = 1  + (−1.29 − 0.0677i)2-s + (−0.686 − 0.848i)3-s + (0.672 + 0.0706i)4-s + (−0.666 − 0.745i)5-s + (0.830 + 1.14i)6-s + (0.318 + 0.948i)7-s + (0.414 + 0.0656i)8-s + (−0.0397 + 0.186i)9-s + (0.811 + 1.00i)10-s + (1.23 − 0.262i)11-s + (−0.401 − 0.618i)12-s + (0.746 + 0.380i)13-s + (−0.346 − 1.24i)14-s + (−0.174 + 1.07i)15-s + (−1.19 − 0.253i)16-s + (0.284 + 0.741i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.185+0.982i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.185+0.982i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.185 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.185+0.982i-0.185 + 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), −0.185+0.982i)(2,\ 175,\ (\ :3/2),\ -0.185 + 0.982i)

Particular Values

L(2)L(2) ≈\approx 0.378484−0.456829i0.378484 - 0.456829i
L(12)L(\frac12) ≈\approx 0.378484−0.456829i0.378484 - 0.456829i
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+(7.45+8.33i)T 1 + (7.45 + 8.33i)T
7 1+(−5.89−17.5i)T 1 + (-5.89 - 17.5i)T
good2 1+(3.65+0.191i)T+(7.95+0.836i)T2 1 + (3.65 + 0.191i)T + (7.95 + 0.836i)T^{2}
3 1+(3.56+4.40i)T+(−5.61+26.4i)T2 1 + (3.56 + 4.40i)T + (-5.61 + 26.4i)T^{2}
11 1+(−45.0+9.57i)T+(1.21e3−541.i)T2 1 + (-45.0 + 9.57i)T + (1.21e3 - 541. i)T^{2}
13 1+(−34.9−17.8i)T+(1.29e3+1.77e3i)T2 1 + (-34.9 - 17.8i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−19.9−51.9i)T+(−3.65e3+3.28e3i)T2 1 + (-19.9 - 51.9i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(3.07+29.2i)T+(−6.70e3+1.42e3i)T2 1 + (3.07 + 29.2i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(−1.05+20.1i)T+(−1.21e4−1.27e3i)T2 1 + (-1.05 + 20.1i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−159.+218.i)T+(−7.53e3−2.31e4i)T2 1 + (-159. + 218. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(−50.0+112.i)T+(−1.99e4−2.21e4i)T2 1 + (-50.0 + 112. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−12.2+7.98i)T+(2.06e4−4.62e4i)T2 1 + (-12.2 + 7.98i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−8.43+2.74i)T+(5.57e4−4.05e4i)T2 1 + (-8.43 + 2.74i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(140.+140.i)T+7.95e4iT2 1 + (140. + 140. i)T + 7.95e4iT^{2}
47 1+(−28.8−11.0i)T+(7.71e4+6.94e4i)T2 1 + (-28.8 - 11.0i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−376.+305.i)T+(3.09e4−1.45e5i)T2 1 + (-376. + 305. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(181.−201.i)T+(−2.14e4−2.04e5i)T2 1 + (181. - 201. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−204.+184.i)T+(2.37e4−2.25e5i)T2 1 + (-204. + 184. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(−332.+127.i)T+(2.23e5−2.01e5i)T2 1 + (-332. + 127. i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(−562.−408.i)T+(1.10e5+3.40e5i)T2 1 + (-562. - 408. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(473.−729.i)T+(−1.58e5−3.55e5i)T2 1 + (473. - 729. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(500.+1.12e3i)T+(−3.29e5+3.66e5i)T2 1 + (500. + 1.12e3i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(110.−699.i)T+(−5.43e5−1.76e5i)T2 1 + (110. - 699. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(368.+409.i)T+(−7.36e4+7.01e5i)T2 1 + (368. + 409. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(193.+1.22e3i)T+(−8.68e5+2.82e5i)T2 1 + (193. + 1.22e3i)T + (-8.68e5 + 2.82e5i)T^{2}
show more
show less
   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

−11.67345903677030602000470471197, −11.32644442786332927576842783226, −9.700699296583003574686203377875, −8.704980351279434287496856677042, −8.215649147852320810140586481138, −6.89917913859799221630545944404, −5.85871352352208008014374521800, −4.21898794463199620947518956340, −1.65058690429350924241670801204, −0.63336229715154040190483129029, 1.05622610383158196512504936737, 3.69940158688977874821531717367, 4.73671838827613333204018347146, 6.59867961872785414921219949651, 7.47286236245063476821324831228, 8.524620130728443113378565716023, 9.755250873058545779333170417421, 10.51986360993042396903117970042, 11.04611427077669010722181158146, 11.93848911306117597130812144048

Graph of the ZZ-function along the critical line