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Properties

Label 2-175-35.9-c3-0-3
Degree 22
Conductor 175175
Sign 0.669−0.742i0.669 - 0.742i
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  + (−2.59 − 1.5i)2-s + (1.73 − i)3-s + (0.5 + 0.866i)4-s − 6·6-s + (−12.1 − 14i)7-s + 21i·8-s + (−11.5 + 19.9i)9-s + (22.5 + 38.9i)11-s + (1.73 + 1.00i)12-s − 59i·13-s + (10.5 + 54.5i)14-s + (35.5 − 61.4i)16-s + (−46.7 + 27i)17-s + (59.7 − 34.5i)18-s + (−60.5 + 104. i)19-s + ⋯
L(s)  = 1  + (−0.918 − 0.530i)2-s + (0.333 − 0.192i)3-s + (0.0625 + 0.108i)4-s − 0.408·6-s + (−0.654 − 0.755i)7-s + 0.928i·8-s + (−0.425 + 0.737i)9-s + (0.616 + 1.06i)11-s + (0.0416 + 0.0240i)12-s − 1.25i·13-s + (0.200 + 1.04i)14-s + (0.554 − 0.960i)16-s + (−0.667 + 0.385i)17-s + (0.782 − 0.451i)18-s + (−0.730 + 1.26i)19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.669−0.742i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.669−0.742i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.669−0.742i0.669 - 0.742i
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ175(149,⋅)\chi_{175} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 0.669−0.742i)(2,\ 175,\ (\ :3/2),\ 0.669 - 0.742i)

Particular Values

L(2)L(2) ≈\approx 0.563868+0.250811i0.563868 + 0.250811i
L(12)L(\frac12) ≈\approx 0.563868+0.250811i0.563868 + 0.250811i
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 1
7 1+(12.1+14i)T 1 + (12.1 + 14i)T
good2 1+(2.59+1.5i)T+(4+6.92i)T2 1 + (2.59 + 1.5i)T + (4 + 6.92i)T^{2}
3 1+(−1.73+i)T+(13.5−23.3i)T2 1 + (-1.73 + i)T + (13.5 - 23.3i)T^{2}
11 1+(−22.5−38.9i)T+(−665.5+1.15e3i)T2 1 + (-22.5 - 38.9i)T + (-665.5 + 1.15e3i)T^{2}
13 1+59iT−2.19e3T2 1 + 59iT - 2.19e3T^{2}
17 1+(46.7−27i)T+(2.45e3−4.25e3i)T2 1 + (46.7 - 27i)T + (2.45e3 - 4.25e3i)T^{2}
19 1+(60.5−104.i)T+(−3.42e3−5.94e3i)T2 1 + (60.5 - 104. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(−59.7−34.5i)T+(6.08e3+1.05e4i)T2 1 + (-59.7 - 34.5i)T + (6.08e3 + 1.05e4i)T^{2}
29 1−162T+2.43e4T2 1 - 162T + 2.43e4T^{2}
31 1+(−44−76.2i)T+(−1.48e4+2.57e4i)T2 1 + (-44 - 76.2i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−224.−129.5i)T+(2.53e4+4.38e4i)T2 1 + (-224. - 129.5i)T + (2.53e4 + 4.38e4i)T^{2}
41 1−195T+6.89e4T2 1 - 195T + 6.89e4T^{2}
43 1−286iT−7.95e4T2 1 - 286iT - 7.95e4T^{2}
47 1+(38.9+22.5i)T+(5.19e4+8.99e4i)T2 1 + (38.9 + 22.5i)T + (5.19e4 + 8.99e4i)T^{2}
53 1+(517.−298.5i)T+(7.44e4−1.28e5i)T2 1 + (517. - 298.5i)T + (7.44e4 - 1.28e5i)T^{2}
59 1+(180+311.i)T+(−1.02e5+1.77e5i)T2 1 + (180 + 311. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(196−339.i)T+(−1.13e5−1.96e5i)T2 1 + (196 - 339. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(242.−140i)T+(1.50e5−2.60e5i)T2 1 + (242. - 140i)T + (1.50e5 - 2.60e5i)T^{2}
71 1−48T+3.57e5T2 1 - 48T + 3.57e5T^{2}
73 1+(578.−334i)T+(1.94e5−3.36e5i)T2 1 + (578. - 334i)T + (1.94e5 - 3.36e5i)T^{2}
79 1+(−391+677.i)T+(−2.46e5−4.26e5i)T2 1 + (-391 + 677. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1+768iT−5.71e5T2 1 + 768iT - 5.71e5T^{2}
89 1+(597−1.03e3i)T+(−3.52e5−6.10e5i)T2 1 + (597 - 1.03e3i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1−902iT−9.12e5T2 1 - 902iT - 9.12e5T^{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

−12.36775377978531296943805471383, −10.97628738449161312161544806072, −10.33057087875985376169367474969, −9.536191286547175278650072720801, −8.372026367714232313728697917052, −7.59963525531400141588979134749, −6.15871435672187466745397182189, −4.58430884525471169418616116709, −2.84725300216554685714252044493, −1.37584895120259891809060741339, 0.39046274048992051658667993163, 2.85057492030130992935031571010, 4.21267453717774901947659681785, 6.26672639852028589878260860366, 6.76872735471695772447324712228, 8.435769808133281364593774830323, 9.089807402808246174352344020339, 9.429621094690656557612325215932, 11.07168381316520743579832822394, 12.03061189006822718553658135072

Graph of the ZZ-function along the critical line