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Properties

Label 2-175-175.17-c1-0-14
Degree 22
Conductor 175175
Sign 0.894+0.447i0.894 + 0.447i
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  + (1.58 − 0.0830i)2-s + (1.23 − 1.52i)3-s + (0.513 − 0.0539i)4-s + (−0.221 + 2.22i)5-s + (1.82 − 2.51i)6-s + (1.63 − 2.08i)7-s + (−2.32 + 0.368i)8-s + (−0.174 − 0.822i)9-s + (−0.166 + 3.54i)10-s + (−2.03 − 0.433i)11-s + (0.550 − 0.848i)12-s + (−1.94 + 0.992i)13-s + (2.40 − 3.43i)14-s + (3.11 + 3.08i)15-s + (−4.66 + 0.990i)16-s + (1.81 − 4.72i)17-s + ⋯
L(s)  = 1  + (1.12 − 0.0587i)2-s + (0.712 − 0.879i)3-s + (0.256 − 0.0269i)4-s + (−0.0992 + 0.995i)5-s + (0.746 − 1.02i)6-s + (0.616 − 0.787i)7-s + (−0.821 + 0.130i)8-s + (−0.0583 − 0.274i)9-s + (−0.0527 + 1.12i)10-s + (−0.614 − 0.130i)11-s + (0.159 − 0.244i)12-s + (−0.540 + 0.275i)13-s + (0.644 − 0.918i)14-s + (0.804 + 0.795i)15-s + (−1.16 + 0.247i)16-s + (0.440 − 1.14i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.894+0.447i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.894+0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.894+0.447i0.894 + 0.447i
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), 0.894+0.447i)(2,\ 175,\ (\ :1/2),\ 0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 2.06754−0.488191i2.06754 - 0.488191i
L(12)L(\frac12) ≈\approx 2.06754−0.488191i2.06754 - 0.488191i
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+(0.221−2.22i)T 1 + (0.221 - 2.22i)T
7 1+(−1.63+2.08i)T 1 + (-1.63 + 2.08i)T
good2 1+(−1.58+0.0830i)T+(1.98−0.209i)T2 1 + (-1.58 + 0.0830i)T + (1.98 - 0.209i)T^{2}
3 1+(−1.23+1.52i)T+(−0.623−2.93i)T2 1 + (-1.23 + 1.52i)T + (-0.623 - 2.93i)T^{2}
11 1+(2.03+0.433i)T+(10.0+4.47i)T2 1 + (2.03 + 0.433i)T + (10.0 + 4.47i)T^{2}
13 1+(1.94−0.992i)T+(7.64−10.5i)T2 1 + (1.94 - 0.992i)T + (7.64 - 10.5i)T^{2}
17 1+(−1.81+4.72i)T+(−12.6−11.3i)T2 1 + (-1.81 + 4.72i)T + (-12.6 - 11.3i)T^{2}
19 1+(0.785−7.47i)T+(−18.5−3.95i)T2 1 + (0.785 - 7.47i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.105+2.00i)T+(−22.8+2.40i)T2 1 + (0.105 + 2.00i)T + (-22.8 + 2.40i)T^{2}
29 1+(4.12+5.67i)T+(−8.96+27.5i)T2 1 + (4.12 + 5.67i)T + (-8.96 + 27.5i)T^{2}
31 1+(−2.58−5.80i)T+(−20.7+23.0i)T2 1 + (-2.58 - 5.80i)T + (-20.7 + 23.0i)T^{2}
37 1+(−2.22−1.44i)T+(15.0+33.8i)T2 1 + (-2.22 - 1.44i)T + (15.0 + 33.8i)T^{2}
41 1+(−8.47−2.75i)T+(33.1+24.0i)T2 1 + (-8.47 - 2.75i)T + (33.1 + 24.0i)T^{2}
43 1+(−0.986+0.986i)T−43iT2 1 + (-0.986 + 0.986i)T - 43iT^{2}
47 1+(−2.45+0.941i)T+(34.9−31.4i)T2 1 + (-2.45 + 0.941i)T + (34.9 - 31.4i)T^{2}
53 1+(7.17+5.81i)T+(11.0+51.8i)T2 1 + (7.17 + 5.81i)T + (11.0 + 51.8i)T^{2}
59 1+(7.04+7.81i)T+(−6.16+58.6i)T2 1 + (7.04 + 7.81i)T + (-6.16 + 58.6i)T^{2}
61 1+(6.37+5.73i)T+(6.37+60.6i)T2 1 + (6.37 + 5.73i)T + (6.37 + 60.6i)T^{2}
67 1+(−3.06−1.17i)T+(49.7+44.8i)T2 1 + (-3.06 - 1.17i)T + (49.7 + 44.8i)T^{2}
71 1+(−5.41+3.93i)T+(21.9−67.5i)T2 1 + (-5.41 + 3.93i)T + (21.9 - 67.5i)T^{2}
73 1+(−4.51−6.94i)T+(−29.6+66.6i)T2 1 + (-4.51 - 6.94i)T + (-29.6 + 66.6i)T^{2}
79 1+(−1.08+2.44i)T+(−52.8−58.7i)T2 1 + (-1.08 + 2.44i)T + (-52.8 - 58.7i)T^{2}
83 1+(−0.456−2.88i)T+(−78.9+25.6i)T2 1 + (-0.456 - 2.88i)T + (-78.9 + 25.6i)T^{2}
89 1+(−8.47+9.41i)T+(−9.30−88.5i)T2 1 + (-8.47 + 9.41i)T + (-9.30 - 88.5i)T^{2}
97 1+(−0.527+3.33i)T+(−92.2−29.9i)T2 1 + (-0.527 + 3.33i)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

−12.86483056118908527971198969749, −11.97782838696010143817251054667, −10.92417511132307293051062785670, −9.755519923195095329525257287568, −8.067797978637905491254319895333, −7.48469996552712524439236298651, −6.30366328508568003738979295598, −4.86100317551990405703741496984, −3.50560017481963541971941452183, −2.33512728686979501436632776764, 2.72049562003849283943005451411, 4.15362055146896126551383314604, 4.91877576455446823708507517277, 5.81985878242921899602193701882, 7.85239884505822452409782861243, 8.970222670478036467386435088947, 9.422911065173177088549765793229, 10.95988286654268609421846912043, 12.24048440223828099289135521290, 12.80322372699200652004422120814

Graph of the ZZ-function along the critical line