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Properties

Label 2-175-175.47-c1-0-10
Degree 22
Conductor 175175
Sign −0.277+0.960i-0.277 + 0.960i
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  + (0.0938 − 1.79i)2-s + (−2.61 + 2.11i)3-s + (−1.21 − 0.127i)4-s + (1.96 − 1.06i)5-s + (3.54 + 4.88i)6-s + (−1.70 − 2.02i)7-s + (0.219 − 1.38i)8-s + (1.72 − 8.13i)9-s + (−1.71 − 3.62i)10-s + (2.72 − 0.579i)11-s + (3.43 − 2.23i)12-s + (0.649 − 1.27i)13-s + (−3.78 + 2.85i)14-s + (−2.89 + 6.94i)15-s + (−4.84 − 1.02i)16-s + (1.22 − 0.470i)17-s + ⋯
L(s)  = 1  + (0.0663 − 1.26i)2-s + (−1.50 + 1.22i)3-s + (−0.605 − 0.0636i)4-s + (0.879 − 0.475i)5-s + (1.44 + 1.99i)6-s + (−0.643 − 0.765i)7-s + (0.0775 − 0.489i)8-s + (0.576 − 2.71i)9-s + (−0.543 − 1.14i)10-s + (0.821 − 0.174i)11-s + (0.992 − 0.644i)12-s + (0.180 − 0.353i)13-s + (−1.01 + 0.763i)14-s + (−0.747 + 1.79i)15-s + (−1.21 − 0.257i)16-s + (0.297 − 0.114i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((−0.277+0.960i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((−0.277+0.960i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: −0.277+0.960i-0.277 + 0.960i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(47,⋅)\chi_{175} (47, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), −0.277+0.960i)(2,\ 175,\ (\ :1/2),\ -0.277 + 0.960i)

Particular Values

L(1)L(1) ≈\approx 0.503023−0.668803i0.503023 - 0.668803i
L(12)L(\frac12) ≈\approx 0.503023−0.668803i0.503023 - 0.668803i
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+(−1.96+1.06i)T 1 + (-1.96 + 1.06i)T
7 1+(1.70+2.02i)T 1 + (1.70 + 2.02i)T
good2 1+(−0.0938+1.79i)T+(−1.98−0.209i)T2 1 + (-0.0938 + 1.79i)T + (-1.98 - 0.209i)T^{2}
3 1+(2.61−2.11i)T+(0.623−2.93i)T2 1 + (2.61 - 2.11i)T + (0.623 - 2.93i)T^{2}
11 1+(−2.72+0.579i)T+(10.0−4.47i)T2 1 + (-2.72 + 0.579i)T + (10.0 - 4.47i)T^{2}
13 1+(−0.649+1.27i)T+(−7.64−10.5i)T2 1 + (-0.649 + 1.27i)T + (-7.64 - 10.5i)T^{2}
17 1+(−1.22+0.470i)T+(12.6−11.3i)T2 1 + (-1.22 + 0.470i)T + (12.6 - 11.3i)T^{2}
19 1+(−0.229−2.18i)T+(−18.5+3.95i)T2 1 + (-0.229 - 2.18i)T + (-18.5 + 3.95i)T^{2}
23 1+(1.79+0.0940i)T+(22.8+2.40i)T2 1 + (1.79 + 0.0940i)T + (22.8 + 2.40i)T^{2}
29 1+(1.22−1.68i)T+(−8.96−27.5i)T2 1 + (1.22 - 1.68i)T + (-8.96 - 27.5i)T^{2}
31 1+(−1.74+3.92i)T+(−20.7−23.0i)T2 1 + (-1.74 + 3.92i)T + (-20.7 - 23.0i)T^{2}
37 1+(−0.0698−0.107i)T+(−15.0+33.8i)T2 1 + (-0.0698 - 0.107i)T + (-15.0 + 33.8i)T^{2}
41 1+(6.08−1.97i)T+(33.1−24.0i)T2 1 + (6.08 - 1.97i)T + (33.1 - 24.0i)T^{2}
43 1+(−2.62+2.62i)T−43iT2 1 + (-2.62 + 2.62i)T - 43iT^{2}
47 1+(2.63−6.85i)T+(−34.9−31.4i)T2 1 + (2.63 - 6.85i)T + (-34.9 - 31.4i)T^{2}
53 1+(−7.79−9.62i)T+(−11.0+51.8i)T2 1 + (-7.79 - 9.62i)T + (-11.0 + 51.8i)T^{2}
59 1+(−0.660+0.733i)T+(−6.16−58.6i)T2 1 + (-0.660 + 0.733i)T + (-6.16 - 58.6i)T^{2}
61 1+(−5.97+5.37i)T+(6.37−60.6i)T2 1 + (-5.97 + 5.37i)T + (6.37 - 60.6i)T^{2}
67 1+(1.09+2.86i)T+(−49.7+44.8i)T2 1 + (1.09 + 2.86i)T + (-49.7 + 44.8i)T^{2}
71 1+(−4.78−3.47i)T+(21.9+67.5i)T2 1 + (-4.78 - 3.47i)T + (21.9 + 67.5i)T^{2}
73 1+(1.03+0.670i)T+(29.6+66.6i)T2 1 + (1.03 + 0.670i)T + (29.6 + 66.6i)T^{2}
79 1+(−3.47−7.80i)T+(−52.8+58.7i)T2 1 + (-3.47 - 7.80i)T + (-52.8 + 58.7i)T^{2}
83 1+(6.21+0.983i)T+(78.9+25.6i)T2 1 + (6.21 + 0.983i)T + (78.9 + 25.6i)T^{2}
89 1+(−2.11−2.35i)T+(−9.30+88.5i)T2 1 + (-2.11 - 2.35i)T + (-9.30 + 88.5i)T^{2}
97 1+(−13.4+2.12i)T+(92.2−29.9i)T2 1 + (-13.4 + 2.12i)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

−12.18140626930253938131332349760, −11.30794093394763684604124797580, −10.42885285488520522941763556090, −9.906274812435908826104823901651, −9.229297016086815082624806379728, −6.66903383128037092845891393129, −5.78348000641277455279608154756, −4.44931498164182704102770907502, −3.52947587407153190165107113627, −0.971893047643615735618022109360, 2.02826508278341169478955474470, 5.15984459977698012903905113082, 5.95445378534834451950638830313, 6.60277862735116365257526998580, 7.15648719095670677464113880848, 8.617066661793233178382058508898, 10.06665287979182304721285912978, 11.32900971061006243139439302120, 12.03604748398632492909989036968, 13.14452607599504347462728458761

Graph of the ZZ-function along the critical line