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Properties

Label 2-175-35.9-c3-0-26
Degree 22
Conductor 175175
Sign 0.391+0.920i0.391 + 0.920i
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  + (1.37 + 0.792i)2-s + (4.54 − 2.62i)3-s + (−2.74 − 4.75i)4-s + 8.31·6-s + (17.8 − 5.10i)7-s − 21.3i·8-s + (0.242 − 0.420i)9-s + (−14.0 − 24.3i)11-s + (−24.9 − 14.3i)12-s + 3.85i·13-s + (28.4 + 7.10i)14-s + (−4.98 + 8.63i)16-s + (33.2 − 19.1i)17-s + (0.666 − 0.384i)18-s + (58.3 − 101. i)19-s + ⋯
L(s)  = 1  + (0.485 + 0.280i)2-s + (0.873 − 0.504i)3-s + (−0.342 − 0.593i)4-s + 0.565·6-s + (0.961 − 0.275i)7-s − 0.945i·8-s + (0.00898 − 0.0155i)9-s + (−0.385 − 0.668i)11-s + (−0.599 − 0.345i)12-s + 0.0823i·13-s + (0.544 + 0.135i)14-s + (−0.0778 + 0.134i)16-s + (0.474 − 0.273i)17-s + (0.00872 − 0.00503i)18-s + (0.704 − 1.22i)19-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.391+0.920i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=((0.391+0.920i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.391+0.920i0.391 + 0.920i
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), 0.391+0.920i)(2,\ 175,\ (\ :3/2),\ 0.391 + 0.920i)

Particular Values

L(2)L(2) ≈\approx 2.27927−1.50657i2.27927 - 1.50657i
L(12)L(\frac12) ≈\approx 2.27927−1.50657i2.27927 - 1.50657i
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+(−17.8+5.10i)T 1 + (-17.8 + 5.10i)T
good2 1+(−1.37−0.792i)T+(4+6.92i)T2 1 + (-1.37 - 0.792i)T + (4 + 6.92i)T^{2}
3 1+(−4.54+2.62i)T+(13.5−23.3i)T2 1 + (-4.54 + 2.62i)T + (13.5 - 23.3i)T^{2}
11 1+(14.0+24.3i)T+(−665.5+1.15e3i)T2 1 + (14.0 + 24.3i)T + (-665.5 + 1.15e3i)T^{2}
13 1−3.85iT−2.19e3T2 1 - 3.85iT - 2.19e3T^{2}
17 1+(−33.2+19.1i)T+(2.45e3−4.25e3i)T2 1 + (-33.2 + 19.1i)T + (2.45e3 - 4.25e3i)T^{2}
19 1+(−58.3+101.i)T+(−3.42e3−5.94e3i)T2 1 + (-58.3 + 101. i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(152.+88.2i)T+(6.08e3+1.05e4i)T2 1 + (152. + 88.2i)T + (6.08e3 + 1.05e4i)T^{2}
29 1−209.T+2.43e4T2 1 - 209.T + 2.43e4T^{2}
31 1+(−103.−179.i)T+(−1.48e4+2.57e4i)T2 1 + (-103. - 179. i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−13.5−7.83i)T+(2.53e4+4.38e4i)T2 1 + (-13.5 - 7.83i)T + (2.53e4 + 4.38e4i)T^{2}
41 1+10.5T+6.89e4T2 1 + 10.5T + 6.89e4T^{2}
43 1−325.iT−7.95e4T2 1 - 325. iT - 7.95e4T^{2}
47 1+(−163.−94.2i)T+(5.19e4+8.99e4i)T2 1 + (-163. - 94.2i)T + (5.19e4 + 8.99e4i)T^{2}
53 1+(−238.+137.i)T+(7.44e4−1.28e5i)T2 1 + (-238. + 137. i)T + (7.44e4 - 1.28e5i)T^{2}
59 1+(−21.9−37.9i)T+(−1.02e5+1.77e5i)T2 1 + (-21.9 - 37.9i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(427.−740.i)T+(−1.13e5−1.96e5i)T2 1 + (427. - 740. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(472.−272.i)T+(1.50e5−2.60e5i)T2 1 + (472. - 272. i)T + (1.50e5 - 2.60e5i)T^{2}
71 1−1.02e3T+3.57e5T2 1 - 1.02e3T + 3.57e5T^{2}
73 1+(−218.+126.i)T+(1.94e5−3.36e5i)T2 1 + (-218. + 126. i)T + (1.94e5 - 3.36e5i)T^{2}
79 1+(461.−799.i)T+(−2.46e5−4.26e5i)T2 1 + (461. - 799. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−960.iT−5.71e5T2 1 - 960. iT - 5.71e5T^{2}
89 1+(66.6−115.i)T+(−3.52e5−6.10e5i)T2 1 + (66.6 - 115. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+1.02e3iT−9.12e5T2 1 + 1.02e3iT - 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

−12.30761325856879038847415801342, −11.05521058695858773422783186979, −10.05502089131653328297126720467, −8.758053226831317174576573880616, −8.002588325499214794026992393198, −6.84996749295681751455424504315, −5.46634607128395574137785395737, −4.45762632590269416334745843534, −2.77984834948785585312412548276, −1.07645672034891283059352257046, 2.17142415664763979969266511036, 3.49317070162148028743768000800, 4.44657389759900781229885196989, 5.66408049851166994678336979841, 7.81849801858679977144026040333, 8.182572900003667010566216623677, 9.381653907995503172218134379889, 10.34819233633807527062174821556, 11.89524539170641555814294146432, 12.19776214090870061611260648517

Graph of the ZZ-function along the critical line