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Properties

Label 2-175-175.17-c1-0-3
Degree 22
Conductor 175175
Sign −0.399−0.916i-0.399 - 0.916i
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  + (−0.0374 + 0.00196i)2-s + (0.0687 − 0.0849i)3-s + (−1.98 + 0.208i)4-s + (−0.633 + 2.14i)5-s + (−0.00240 + 0.00331i)6-s + (−2.20 + 1.46i)7-s + (0.147 − 0.0234i)8-s + (0.621 + 2.92i)9-s + (0.0195 − 0.0814i)10-s + (−1.34 − 0.286i)11-s + (−0.118 + 0.183i)12-s + (1.43 − 0.730i)13-s + (0.0795 − 0.0591i)14-s + (0.138 + 0.201i)15-s + (3.90 − 0.829i)16-s + (−1.56 + 4.07i)17-s + ⋯
L(s)  = 1  + (−0.0264 + 0.00138i)2-s + (0.0397 − 0.0490i)3-s + (−0.993 + 0.104i)4-s + (−0.283 + 0.959i)5-s + (−0.000982 + 0.00135i)6-s + (−0.832 + 0.553i)7-s + (0.0523 − 0.00828i)8-s + (0.207 + 0.974i)9-s + (0.00616 − 0.0257i)10-s + (−0.405 − 0.0862i)11-s + (−0.0343 + 0.0528i)12-s + (0.397 − 0.202i)13-s + (0.0212 − 0.0158i)14-s + (0.0357 + 0.0519i)15-s + (0.976 − 0.207i)16-s + (−0.379 + 0.988i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.399−0.916i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.399−0.916i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.399−0.916i-0.399 - 0.916i
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), −0.399−0.916i)(2,\ 175,\ (\ :1/2),\ -0.399 - 0.916i)

Particular Values

L(1)L(1) ≈\approx 0.363124+0.554380i0.363124 + 0.554380i
L(12)L(\frac12) ≈\approx 0.363124+0.554380i0.363124 + 0.554380i
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+(0.633−2.14i)T 1 + (0.633 - 2.14i)T
7 1+(2.20−1.46i)T 1 + (2.20 - 1.46i)T
good2 1+(0.0374−0.00196i)T+(1.98−0.209i)T2 1 + (0.0374 - 0.00196i)T + (1.98 - 0.209i)T^{2}
3 1+(−0.0687+0.0849i)T+(−0.623−2.93i)T2 1 + (-0.0687 + 0.0849i)T + (-0.623 - 2.93i)T^{2}
11 1+(1.34+0.286i)T+(10.0+4.47i)T2 1 + (1.34 + 0.286i)T + (10.0 + 4.47i)T^{2}
13 1+(−1.43+0.730i)T+(7.64−10.5i)T2 1 + (-1.43 + 0.730i)T + (7.64 - 10.5i)T^{2}
17 1+(1.56−4.07i)T+(−12.6−11.3i)T2 1 + (1.56 - 4.07i)T + (-12.6 - 11.3i)T^{2}
19 1+(0.00716−0.0681i)T+(−18.5−3.95i)T2 1 + (0.00716 - 0.0681i)T + (-18.5 - 3.95i)T^{2}
23 1+(0.295+5.63i)T+(−22.8+2.40i)T2 1 + (0.295 + 5.63i)T + (-22.8 + 2.40i)T^{2}
29 1+(−3.88−5.35i)T+(−8.96+27.5i)T2 1 + (-3.88 - 5.35i)T + (-8.96 + 27.5i)T^{2}
31 1+(−1.42−3.20i)T+(−20.7+23.0i)T2 1 + (-1.42 - 3.20i)T + (-20.7 + 23.0i)T^{2}
37 1+(6.49+4.21i)T+(15.0+33.8i)T2 1 + (6.49 + 4.21i)T + (15.0 + 33.8i)T^{2}
41 1+(−9.12−2.96i)T+(33.1+24.0i)T2 1 + (-9.12 - 2.96i)T + (33.1 + 24.0i)T^{2}
43 1+(3.38−3.38i)T−43iT2 1 + (3.38 - 3.38i)T - 43iT^{2}
47 1+(−6.65+2.55i)T+(34.9−31.4i)T2 1 + (-6.65 + 2.55i)T + (34.9 - 31.4i)T^{2}
53 1+(−2.30−1.86i)T+(11.0+51.8i)T2 1 + (-2.30 - 1.86i)T + (11.0 + 51.8i)T^{2}
59 1+(−6.88−7.64i)T+(−6.16+58.6i)T2 1 + (-6.88 - 7.64i)T + (-6.16 + 58.6i)T^{2}
61 1+(−5.62−5.06i)T+(6.37+60.6i)T2 1 + (-5.62 - 5.06i)T + (6.37 + 60.6i)T^{2}
67 1+(8.28+3.18i)T+(49.7+44.8i)T2 1 + (8.28 + 3.18i)T + (49.7 + 44.8i)T^{2}
71 1+(−5.52+4.01i)T+(21.9−67.5i)T2 1 + (-5.52 + 4.01i)T + (21.9 - 67.5i)T^{2}
73 1+(0.233+0.359i)T+(−29.6+66.6i)T2 1 + (0.233 + 0.359i)T + (-29.6 + 66.6i)T^{2}
79 1+(3.70−8.31i)T+(−52.8−58.7i)T2 1 + (3.70 - 8.31i)T + (-52.8 - 58.7i)T^{2}
83 1+(−0.663−4.19i)T+(−78.9+25.6i)T2 1 + (-0.663 - 4.19i)T + (-78.9 + 25.6i)T^{2}
89 1+(5.38−5.98i)T+(−9.30−88.5i)T2 1 + (5.38 - 5.98i)T + (-9.30 - 88.5i)T^{2}
97 1+(2.59−16.3i)T+(−92.2−29.9i)T2 1 + (2.59 - 16.3i)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

−13.03569777353592976958235360381, −12.29543031812082675078667323581, −10.69347390343842009443653002934, −10.27321365907233995003572647732, −8.863510950414942924213547324843, −8.065082955084450673570878848469, −6.76310603703295824664423613654, −5.53201649285790478084630093346, −4.08519572112683341007534737314, −2.74816327879635601932379705399, 0.64254826302408406250477063436, 3.60048323489881482859181013965, 4.53897413015696328973454777320, 5.84211148937111581582984268644, 7.28469081363620519026147377990, 8.556655442421699999116852303421, 9.418201583989384896100347577722, 10.00595760663901262319690503297, 11.61963013928648844441099675043, 12.59041458733463643186447079594

Graph of the ZZ-function along the critical line