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Properties

Label 2-175-175.108-c1-0-16
Degree 22
Conductor 175175
Sign −0.567−0.823i-0.567 - 0.823i
Analytic cond. 1.397381.39738
Root an. cond. 1.182101.18210
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.139 − 2.65i)2-s + (0.171 + 0.138i)3-s + (−5.03 + 0.528i)4-s + (−2.22 − 0.174i)5-s + (0.344 − 0.474i)6-s + (−2.47 + 0.932i)7-s + (1.27 + 8.02i)8-s + (−0.613 − 2.88i)9-s + (−0.151 + 5.93i)10-s + (2.92 + 0.621i)11-s + (−0.936 − 0.608i)12-s + (−2.23 − 4.38i)13-s + (2.81 + 6.43i)14-s + (−0.358 − 0.339i)15-s + (11.2 − 2.38i)16-s + (−4.48 − 1.72i)17-s + ⋯
L(s)  = 1  + (−0.0983 − 1.87i)2-s + (0.0990 + 0.0802i)3-s + (−2.51 + 0.264i)4-s + (−0.996 − 0.0778i)5-s + (0.140 − 0.193i)6-s + (−0.935 + 0.352i)7-s + (0.449 + 2.83i)8-s + (−0.204 − 0.962i)9-s + (−0.0480 + 1.87i)10-s + (0.882 + 0.187i)11-s + (−0.270 − 0.175i)12-s + (−0.619 − 1.21i)13-s + (0.753 + 1.72i)14-s + (−0.0925 − 0.0877i)15-s + (2.80 − 0.596i)16-s + (−1.08 − 0.417i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.567−0.823i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.567−0.823i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.567−0.823i-0.567 - 0.823i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(108,⋅)\chi_{175} (108, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.567−0.823i)(2,\ 175,\ (\ :1/2),\ -0.567 - 0.823i)

Particular Values

L(1)L(1) ≈\approx 0.218964+0.416748i0.218964 + 0.416748i
L(12)L(\frac12) ≈\approx 0.218964+0.416748i0.218964 + 0.416748i
L(32)L(\frac{3}{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+(2.22+0.174i)T 1 + (2.22 + 0.174i)T
7 1+(2.47−0.932i)T 1 + (2.47 - 0.932i)T
good2 1+(0.139+2.65i)T+(−1.98+0.209i)T2 1 + (0.139 + 2.65i)T + (-1.98 + 0.209i)T^{2}
3 1+(−0.171−0.138i)T+(0.623+2.93i)T2 1 + (-0.171 - 0.138i)T + (0.623 + 2.93i)T^{2}
11 1+(−2.92−0.621i)T+(10.0+4.47i)T2 1 + (-2.92 - 0.621i)T + (10.0 + 4.47i)T^{2}
13 1+(2.23+4.38i)T+(−7.64+10.5i)T2 1 + (2.23 + 4.38i)T + (-7.64 + 10.5i)T^{2}
17 1+(4.48+1.72i)T+(12.6+11.3i)T2 1 + (4.48 + 1.72i)T + (12.6 + 11.3i)T^{2}
19 1+(−0.376+3.58i)T+(−18.5−3.95i)T2 1 + (-0.376 + 3.58i)T + (-18.5 - 3.95i)T^{2}
23 1+(−3.26+0.171i)T+(22.8−2.40i)T2 1 + (-3.26 + 0.171i)T + (22.8 - 2.40i)T^{2}
29 1+(−2.87−3.95i)T+(−8.96+27.5i)T2 1 + (-2.87 - 3.95i)T + (-8.96 + 27.5i)T^{2}
31 1+(0.657+1.47i)T+(−20.7+23.0i)T2 1 + (0.657 + 1.47i)T + (-20.7 + 23.0i)T^{2}
37 1+(0.731−1.12i)T+(−15.0−33.8i)T2 1 + (0.731 - 1.12i)T + (-15.0 - 33.8i)T^{2}
41 1+(4.99+1.62i)T+(33.1+24.0i)T2 1 + (4.99 + 1.62i)T + (33.1 + 24.0i)T^{2}
43 1+(8.43+8.43i)T+43iT2 1 + (8.43 + 8.43i)T + 43iT^{2}
47 1+(−1.19−3.12i)T+(−34.9+31.4i)T2 1 + (-1.19 - 3.12i)T + (-34.9 + 31.4i)T^{2}
53 1+(2.69−3.32i)T+(−11.0−51.8i)T2 1 + (2.69 - 3.32i)T + (-11.0 - 51.8i)T^{2}
59 1+(−0.374−0.415i)T+(−6.16+58.6i)T2 1 + (-0.374 - 0.415i)T + (-6.16 + 58.6i)T^{2}
61 1+(−7.43−6.69i)T+(6.37+60.6i)T2 1 + (-7.43 - 6.69i)T + (6.37 + 60.6i)T^{2}
67 1+(−3.34+8.72i)T+(−49.7−44.8i)T2 1 + (-3.34 + 8.72i)T + (-49.7 - 44.8i)T^{2}
71 1+(0.293−0.213i)T+(21.9−67.5i)T2 1 + (0.293 - 0.213i)T + (21.9 - 67.5i)T^{2}
73 1+(−2.45+1.59i)T+(29.6−66.6i)T2 1 + (-2.45 + 1.59i)T + (29.6 - 66.6i)T^{2}
79 1+(−2.12+4.78i)T+(−52.8−58.7i)T2 1 + (-2.12 + 4.78i)T + (-52.8 - 58.7i)T^{2}
83 1+(−0.898+0.142i)T+(78.9−25.6i)T2 1 + (-0.898 + 0.142i)T + (78.9 - 25.6i)T^{2}
89 1+(−6.04+6.71i)T+(−9.30−88.5i)T2 1 + (-6.04 + 6.71i)T + (-9.30 - 88.5i)T^{2}
97 1+(7.15+1.13i)T+(92.2+29.9i)T2 1 + (7.15 + 1.13i)T + (92.2 + 29.9i)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

−12.06442970372479401720974236634, −11.27070755009581143628473958670, −10.19862066653043484537976963160, −9.183781432377386430403545323056, −8.670024925150168209743659243146, −6.89103029380840808676225912738, −4.90237042244856720215718131398, −3.63822491746649417033944826673, −2.86285808257194137237245644463, −0.44902710593545335940279555118, 3.86858844899234535106369774907, 4.83880808179649263355166685997, 6.48219815618299220859706670457, 6.97143323894220805604031184201, 8.071197409800607266779765239071, 8.880100211670605108141248506332, 9.941845314424383719772536076111, 11.44796454906540462850064741672, 12.78814721134021617573346612073, 13.72363234649486315006859133321

Graph of the ZZ-function along the critical line