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Properties

Label 2-175-175.103-c1-0-3
Degree 22
Conductor 175175
Sign 0.336−0.941i0.336 - 0.941i
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  + (−2.77 − 0.145i)2-s + (0.835 + 1.03i)3-s + (5.68 + 0.597i)4-s + (0.745 + 2.10i)5-s + (−2.16 − 2.98i)6-s + (2.14 + 1.54i)7-s + (−10.2 − 1.61i)8-s + (0.257 − 1.21i)9-s + (−1.76 − 5.95i)10-s + (0.0303 − 0.00645i)11-s + (4.13 + 6.36i)12-s + (−2.44 − 1.24i)13-s + (−5.73 − 4.60i)14-s + (−1.55 + 2.52i)15-s + (16.9 + 3.59i)16-s + (1.52 + 3.97i)17-s + ⋯
L(s)  = 1  + (−1.96 − 0.102i)2-s + (0.482 + 0.595i)3-s + (2.84 + 0.298i)4-s + (0.333 + 0.942i)5-s + (−0.884 − 1.21i)6-s + (0.811 + 0.584i)7-s + (−3.61 − 0.571i)8-s + (0.0859 − 0.404i)9-s + (−0.557 − 1.88i)10-s + (0.00916 − 0.00194i)11-s + (1.19 + 1.83i)12-s + (−0.676 − 0.344i)13-s + (−1.53 − 1.23i)14-s + (−0.400 + 0.653i)15-s + (4.22 + 0.898i)16-s + (0.369 + 0.963i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.336−0.941i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.336−0.941i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.336−0.941i0.336 - 0.941i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(103,⋅)\chi_{175} (103, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.336−0.941i)(2,\ 175,\ (\ :1/2),\ 0.336 - 0.941i)

Particular Values

L(1)L(1) ≈\approx 0.534813+0.376902i0.534813 + 0.376902i
L(12)L(\frac12) ≈\approx 0.534813+0.376902i0.534813 + 0.376902i
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+(−0.745−2.10i)T 1 + (-0.745 - 2.10i)T
7 1+(−2.14−1.54i)T 1 + (-2.14 - 1.54i)T
good2 1+(2.77+0.145i)T+(1.98+0.209i)T2 1 + (2.77 + 0.145i)T + (1.98 + 0.209i)T^{2}
3 1+(−0.835−1.03i)T+(−0.623+2.93i)T2 1 + (-0.835 - 1.03i)T + (-0.623 + 2.93i)T^{2}
11 1+(−0.0303+0.00645i)T+(10.0−4.47i)T2 1 + (-0.0303 + 0.00645i)T + (10.0 - 4.47i)T^{2}
13 1+(2.44+1.24i)T+(7.64+10.5i)T2 1 + (2.44 + 1.24i)T + (7.64 + 10.5i)T^{2}
17 1+(−1.52−3.97i)T+(−12.6+11.3i)T2 1 + (-1.52 - 3.97i)T + (-12.6 + 11.3i)T^{2}
19 1+(−0.275−2.62i)T+(−18.5+3.95i)T2 1 + (-0.275 - 2.62i)T + (-18.5 + 3.95i)T^{2}
23 1+(−0.103+1.97i)T+(−22.8−2.40i)T2 1 + (-0.103 + 1.97i)T + (-22.8 - 2.40i)T^{2}
29 1+(−0.472+0.649i)T+(−8.96−27.5i)T2 1 + (-0.472 + 0.649i)T + (-8.96 - 27.5i)T^{2}
31 1+(0.468−1.05i)T+(−20.7−23.0i)T2 1 + (0.468 - 1.05i)T + (-20.7 - 23.0i)T^{2}
37 1+(0.394−0.255i)T+(15.0−33.8i)T2 1 + (0.394 - 0.255i)T + (15.0 - 33.8i)T^{2}
41 1+(3.35−1.08i)T+(33.1−24.0i)T2 1 + (3.35 - 1.08i)T + (33.1 - 24.0i)T^{2}
43 1+(6.90+6.90i)T+43iT2 1 + (6.90 + 6.90i)T + 43iT^{2}
47 1+(−3.24−1.24i)T+(34.9+31.4i)T2 1 + (-3.24 - 1.24i)T + (34.9 + 31.4i)T^{2}
53 1+(−8.21+6.64i)T+(11.0−51.8i)T2 1 + (-8.21 + 6.64i)T + (11.0 - 51.8i)T^{2}
59 1+(1.58−1.76i)T+(−6.16−58.6i)T2 1 + (1.58 - 1.76i)T + (-6.16 - 58.6i)T^{2}
61 1+(−0.651+0.586i)T+(6.37−60.6i)T2 1 + (-0.651 + 0.586i)T + (6.37 - 60.6i)T^{2}
67 1+(−9.68+3.71i)T+(49.7−44.8i)T2 1 + (-9.68 + 3.71i)T + (49.7 - 44.8i)T^{2}
71 1+(2.26+1.64i)T+(21.9+67.5i)T2 1 + (2.26 + 1.64i)T + (21.9 + 67.5i)T^{2}
73 1+(3.90−6.01i)T+(−29.6−66.6i)T2 1 + (3.90 - 6.01i)T + (-29.6 - 66.6i)T^{2}
79 1+(4.45+10.0i)T+(−52.8+58.7i)T2 1 + (4.45 + 10.0i)T + (-52.8 + 58.7i)T^{2}
83 1+(−2.29+14.4i)T+(−78.9−25.6i)T2 1 + (-2.29 + 14.4i)T + (-78.9 - 25.6i)T^{2}
89 1+(3.12+3.47i)T+(−9.30+88.5i)T2 1 + (3.12 + 3.47i)T + (-9.30 + 88.5i)T^{2}
97 1+(0.0409+0.258i)T+(−92.2+29.9i)T2 1 + (0.0409 + 0.258i)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

−12.33446291272637355374742878264, −11.50478799019058739050846016853, −10.34513937777781560078366294062, −10.03898966374833192484844474056, −8.894122801912340820023780169833, −8.148345980217556698073065735780, −7.07050655643058493717124402762, −5.90609638434582329155024536074, −3.27287572252093809388562587666, −1.99979752727351560993746114426, 1.16776903696735866588507916619, 2.35104441945601269227298936720, 5.17496070417930013042731893805, 6.93373545533882730342917641387, 7.66170093115102767857877659012, 8.422125800463690273469651827465, 9.319920967855653799180555994500, 10.19126641225114426706819427904, 11.31836680666294505633080160934, 12.15316120756119363412196105777

Graph of the ZZ-function along the critical line