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Properties

Label 2-175-175.17-c1-0-6
Degree 22
Conductor 175175
Sign 0.584−0.811i0.584 - 0.811i
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  + (0.776 − 0.0406i)2-s + (−0.706 + 0.872i)3-s + (−1.38 + 0.145i)4-s + (2.15 + 0.589i)5-s + (−0.513 + 0.706i)6-s + (2.33 + 1.24i)7-s + (−2.60 + 0.412i)8-s + (0.361 + 1.70i)9-s + (1.69 + 0.370i)10-s + (1.29 + 0.276i)11-s + (0.853 − 1.31i)12-s + (−1.79 + 0.912i)13-s + (1.86 + 0.870i)14-s + (−2.03 + 1.46i)15-s + (0.724 − 0.153i)16-s + (0.333 − 0.868i)17-s + ⋯
L(s)  = 1  + (0.548 − 0.0287i)2-s + (−0.408 + 0.503i)3-s + (−0.694 + 0.0729i)4-s + (0.964 + 0.263i)5-s + (−0.209 + 0.288i)6-s + (0.882 + 0.470i)7-s + (−0.921 + 0.145i)8-s + (0.120 + 0.566i)9-s + (0.536 + 0.117i)10-s + (0.391 + 0.0832i)11-s + (0.246 − 0.379i)12-s + (−0.496 + 0.253i)13-s + (0.497 + 0.232i)14-s + (−0.526 + 0.378i)15-s + (0.181 − 0.0384i)16-s + (0.0808 − 0.210i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.584−0.811i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.584−0.811i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.584−0.811i0.584 - 0.811i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(17,⋅)\chi_{175} (17, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.584−0.811i)(2,\ 175,\ (\ :1/2),\ 0.584 - 0.811i)

Particular Values

L(1)L(1) ≈\approx 1.19656+0.612623i1.19656 + 0.612623i
L(12)L(\frac12) ≈\approx 1.19656+0.612623i1.19656 + 0.612623i
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+(−2.15−0.589i)T 1 + (-2.15 - 0.589i)T
7 1+(−2.33−1.24i)T 1 + (-2.33 - 1.24i)T
good2 1+(−0.776+0.0406i)T+(1.98−0.209i)T2 1 + (-0.776 + 0.0406i)T + (1.98 - 0.209i)T^{2}
3 1+(0.706−0.872i)T+(−0.623−2.93i)T2 1 + (0.706 - 0.872i)T + (-0.623 - 2.93i)T^{2}
11 1+(−1.29−0.276i)T+(10.0+4.47i)T2 1 + (-1.29 - 0.276i)T + (10.0 + 4.47i)T^{2}
13 1+(1.79−0.912i)T+(7.64−10.5i)T2 1 + (1.79 - 0.912i)T + (7.64 - 10.5i)T^{2}
17 1+(−0.333+0.868i)T+(−12.6−11.3i)T2 1 + (-0.333 + 0.868i)T + (-12.6 - 11.3i)T^{2}
19 1+(−0.550+5.24i)T+(−18.5−3.95i)T2 1 + (-0.550 + 5.24i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.0599+1.14i)T+(−22.8+2.40i)T2 1 + (0.0599 + 1.14i)T + (-22.8 + 2.40i)T^{2}
29 1+(4.04+5.56i)T+(−8.96+27.5i)T2 1 + (4.04 + 5.56i)T + (-8.96 + 27.5i)T^{2}
31 1+(3.98+8.95i)T+(−20.7+23.0i)T2 1 + (3.98 + 8.95i)T + (-20.7 + 23.0i)T^{2}
37 1+(−6.42−4.17i)T+(15.0+33.8i)T2 1 + (-6.42 - 4.17i)T + (15.0 + 33.8i)T^{2}
41 1+(0.0622+0.0202i)T+(33.1+24.0i)T2 1 + (0.0622 + 0.0202i)T + (33.1 + 24.0i)T^{2}
43 1+(3.34−3.34i)T−43iT2 1 + (3.34 - 3.34i)T - 43iT^{2}
47 1+(0.174−0.0668i)T+(34.9−31.4i)T2 1 + (0.174 - 0.0668i)T + (34.9 - 31.4i)T^{2}
53 1+(−8.68−7.02i)T+(11.0+51.8i)T2 1 + (-8.68 - 7.02i)T + (11.0 + 51.8i)T^{2}
59 1+(3.22+3.58i)T+(−6.16+58.6i)T2 1 + (3.22 + 3.58i)T + (-6.16 + 58.6i)T^{2}
61 1+(3.39+3.06i)T+(6.37+60.6i)T2 1 + (3.39 + 3.06i)T + (6.37 + 60.6i)T^{2}
67 1+(7.93+3.04i)T+(49.7+44.8i)T2 1 + (7.93 + 3.04i)T + (49.7 + 44.8i)T^{2}
71 1+(−8.72+6.33i)T+(21.9−67.5i)T2 1 + (-8.72 + 6.33i)T + (21.9 - 67.5i)T^{2}
73 1+(−0.896−1.38i)T+(−29.6+66.6i)T2 1 + (-0.896 - 1.38i)T + (-29.6 + 66.6i)T^{2}
79 1+(4.24−9.53i)T+(−52.8−58.7i)T2 1 + (4.24 - 9.53i)T + (-52.8 - 58.7i)T^{2}
83 1+(0.978+6.17i)T+(−78.9+25.6i)T2 1 + (0.978 + 6.17i)T + (-78.9 + 25.6i)T^{2}
89 1+(−10.5+11.6i)T+(−9.30−88.5i)T2 1 + (-10.5 + 11.6i)T + (-9.30 - 88.5i)T^{2}
97 1+(2.72−17.1i)T+(−92.2−29.9i)T2 1 + (2.72 - 17.1i)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

−13.13535052281042856613941264830, −11.80636030565306054729198371560, −11.03490433071029955162187331960, −9.742641477478632480294266508654, −9.141274687307035200843225384365, −7.73777245847085111026012866771, −6.06909541362419225635690980822, −5.14593767640431741675987249336, −4.38026632458168829001970546952, −2.39876313751571396058093917783, 1.39581107314121545323264695282, 3.74816270273198031965414829110, 5.13309211378218191923573553454, 5.88886730756046023360283201285, 7.15851068197433543653260358034, 8.587319989341199460029688651157, 9.546783522798656373345761275108, 10.57492757490662908961872340463, 11.97435371221793579220665047369, 12.64335573193334021234865419090

Graph of the ZZ-function along the critical line