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Properties

Label 2-175-175.103-c3-0-19
Degree 22
Conductor 175175
Sign −0.602−0.798i-0.602 - 0.798i
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  + (2.29 + 0.120i)2-s + (5.30 + 6.55i)3-s + (−2.70 − 0.284i)4-s + (−8.16 + 7.63i)5-s + (11.3 + 15.6i)6-s + (16.6 + 8.19i)7-s + (−24.3 − 3.85i)8-s + (−9.17 + 43.1i)9-s + (−19.6 + 16.5i)10-s + (38.5 − 8.19i)11-s + (−12.4 − 19.2i)12-s + (−76.5 − 39.0i)13-s + (37.1 + 20.8i)14-s + (−93.3 − 12.9i)15-s + (−34.1 − 7.24i)16-s + (36.9 + 96.2i)17-s + ⋯
L(s)  = 1  + (0.811 + 0.0425i)2-s + (1.02 + 1.26i)3-s + (−0.337 − 0.0355i)4-s + (−0.730 + 0.683i)5-s + (0.775 + 1.06i)6-s + (0.896 + 0.442i)7-s + (−1.07 − 0.170i)8-s + (−0.339 + 1.59i)9-s + (−0.621 + 0.523i)10-s + (1.05 − 0.224i)11-s + (−0.300 − 0.462i)12-s + (−1.63 − 0.832i)13-s + (0.708 + 0.397i)14-s + (−1.60 − 0.223i)15-s + (−0.532 − 0.113i)16-s + (0.527 + 1.37i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.602−0.798i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((−0.602−0.798i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.602−0.798i-0.602 - 0.798i
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), −0.602−0.798i)(2,\ 175,\ (\ :3/2),\ -0.602 - 0.798i)

Particular Values

L(2)L(2) ≈\approx 1.18329+2.37475i1.18329 + 2.37475i
L(12)L(\frac12) ≈\approx 1.18329+2.37475i1.18329 + 2.37475i
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+(8.16−7.63i)T 1 + (8.16 - 7.63i)T
7 1+(−16.6−8.19i)T 1 + (-16.6 - 8.19i)T
good2 1+(−2.29−0.120i)T+(7.95+0.836i)T2 1 + (-2.29 - 0.120i)T + (7.95 + 0.836i)T^{2}
3 1+(−5.30−6.55i)T+(−5.61+26.4i)T2 1 + (-5.30 - 6.55i)T + (-5.61 + 26.4i)T^{2}
11 1+(−38.5+8.19i)T+(1.21e3−541.i)T2 1 + (-38.5 + 8.19i)T + (1.21e3 - 541. i)T^{2}
13 1+(76.5+39.0i)T+(1.29e3+1.77e3i)T2 1 + (76.5 + 39.0i)T + (1.29e3 + 1.77e3i)T^{2}
17 1+(−36.9−96.2i)T+(−3.65e3+3.28e3i)T2 1 + (-36.9 - 96.2i)T + (-3.65e3 + 3.28e3i)T^{2}
19 1+(−6.01−57.2i)T+(−6.70e3+1.42e3i)T2 1 + (-6.01 - 57.2i)T + (-6.70e3 + 1.42e3i)T^{2}
23 1+(2.32−44.4i)T+(−1.21e4−1.27e3i)T2 1 + (2.32 - 44.4i)T + (-1.21e4 - 1.27e3i)T^{2}
29 1+(−128.+177.i)T+(−7.53e3−2.31e4i)T2 1 + (-128. + 177. i)T + (-7.53e3 - 2.31e4i)T^{2}
31 1+(76.2−171.i)T+(−1.99e4−2.21e4i)T2 1 + (76.2 - 171. i)T + (-1.99e4 - 2.21e4i)T^{2}
37 1+(−237.+154.i)T+(2.06e4−4.62e4i)T2 1 + (-237. + 154. i)T + (2.06e4 - 4.62e4i)T^{2}
41 1+(−8.31+2.70i)T+(5.57e4−4.05e4i)T2 1 + (-8.31 + 2.70i)T + (5.57e4 - 4.05e4i)T^{2}
43 1+(−233.−233.i)T+7.95e4iT2 1 + (-233. - 233. i)T + 7.95e4iT^{2}
47 1+(−183.−70.3i)T+(7.71e4+6.94e4i)T2 1 + (-183. - 70.3i)T + (7.71e4 + 6.94e4i)T^{2}
53 1+(−231.+187.i)T+(3.09e4−1.45e5i)T2 1 + (-231. + 187. i)T + (3.09e4 - 1.45e5i)T^{2}
59 1+(119.−132.i)T+(−2.14e4−2.04e5i)T2 1 + (119. - 132. i)T + (-2.14e4 - 2.04e5i)T^{2}
61 1+(−336.+302.i)T+(2.37e4−2.25e5i)T2 1 + (-336. + 302. i)T + (2.37e4 - 2.25e5i)T^{2}
67 1+(199.−76.5i)T+(2.23e5−2.01e5i)T2 1 + (199. - 76.5i)T + (2.23e5 - 2.01e5i)T^{2}
71 1+(571.+415.i)T+(1.10e5+3.40e5i)T2 1 + (571. + 415. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(90.8−139.i)T+(−1.58e5−3.55e5i)T2 1 + (90.8 - 139. i)T + (-1.58e5 - 3.55e5i)T^{2}
79 1+(−21.6−48.6i)T+(−3.29e5+3.66e5i)T2 1 + (-21.6 - 48.6i)T + (-3.29e5 + 3.66e5i)T^{2}
83 1+(41.9−264.i)T+(−5.43e5−1.76e5i)T2 1 + (41.9 - 264. i)T + (-5.43e5 - 1.76e5i)T^{2}
89 1+(802.+890.i)T+(−7.36e4+7.01e5i)T2 1 + (802. + 890. i)T + (-7.36e4 + 7.01e5i)T^{2}
97 1+(54.9+347.i)T+(−8.68e5+2.82e5i)T2 1 + (54.9 + 347. 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

−12.57097627598695150045708046244, −11.76234359068180635468915480378, −10.47610988318368287741060088026, −9.623839009652425118412700015350, −8.540872351935597022399942245207, −7.75181773707065985729433981260, −5.81061106446934075722156278307, −4.55342114704115067348453702349, −3.83663902178648595522085889125, −2.78115784032354832958710315398, 0.893045437498382894921775915518, 2.58879575173652230953393225885, 4.14474945462937653203606327755, 4.98758502191398690998522691125, 6.98222914058673202402868972383, 7.59852171287831508401237874929, 8.774233448857328322007941498934, 9.386234247109621858147374212428, 11.79213779265742166040205590314, 11.98308673251733451153270750221

Graph of the ZZ-function along the critical line