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Properties

Label 2-175-175.108-c1-0-12
Degree 22
Conductor 175175
Sign 0.699+0.714i0.699 + 0.714i
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  + (−0.0729 − 1.39i)2-s + (2.18 + 1.77i)3-s + (0.0566 − 0.00595i)4-s + (−0.287 − 2.21i)5-s + (2.30 − 3.17i)6-s + (−2.62 + 0.311i)7-s + (−0.448 − 2.83i)8-s + (1.02 + 4.82i)9-s + (−3.06 + 0.562i)10-s + (1.26 + 0.268i)11-s + (0.134 + 0.0873i)12-s + (1.23 + 2.42i)13-s + (0.624 + 3.63i)14-s + (3.30 − 5.36i)15-s + (−3.79 + 0.807i)16-s + (1.66 + 0.639i)17-s + ⋯
L(s)  = 1  + (−0.0515 − 0.984i)2-s + (1.26 + 1.02i)3-s + (0.0283 − 0.00297i)4-s + (−0.128 − 0.991i)5-s + (0.942 − 1.29i)6-s + (−0.993 + 0.117i)7-s + (−0.158 − 1.00i)8-s + (0.341 + 1.60i)9-s + (−0.969 + 0.177i)10-s + (0.380 + 0.0808i)11-s + (0.0388 + 0.0252i)12-s + (0.342 + 0.672i)13-s + (0.167 + 0.971i)14-s + (0.852 − 1.38i)15-s + (−0.949 + 0.201i)16-s + (0.403 + 0.155i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.699+0.714i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.699+0.714i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.699+0.714i0.699 + 0.714i
Analytic conductor: 1.397381.39738
Root analytic conductor: 1.182101.18210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ175(108,⋅)\chi_{175} (108, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 175, ( :1/2), 0.699+0.714i)(2,\ 175,\ (\ :1/2),\ 0.699 + 0.714i)

Particular Values

L(1)L(1) ≈\approx 1.47935−0.621663i1.47935 - 0.621663i
L(12)L(\frac12) ≈\approx 1.47935−0.621663i1.47935 - 0.621663i
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+(0.287+2.21i)T 1 + (0.287 + 2.21i)T
7 1+(2.62−0.311i)T 1 + (2.62 - 0.311i)T
good2 1+(0.0729+1.39i)T+(−1.98+0.209i)T2 1 + (0.0729 + 1.39i)T + (-1.98 + 0.209i)T^{2}
3 1+(−2.18−1.77i)T+(0.623+2.93i)T2 1 + (-2.18 - 1.77i)T + (0.623 + 2.93i)T^{2}
11 1+(−1.26−0.268i)T+(10.0+4.47i)T2 1 + (-1.26 - 0.268i)T + (10.0 + 4.47i)T^{2}
13 1+(−1.23−2.42i)T+(−7.64+10.5i)T2 1 + (-1.23 - 2.42i)T + (-7.64 + 10.5i)T^{2}
17 1+(−1.66−0.639i)T+(12.6+11.3i)T2 1 + (-1.66 - 0.639i)T + (12.6 + 11.3i)T^{2}
19 1+(0.525−4.99i)T+(−18.5−3.95i)T2 1 + (0.525 - 4.99i)T + (-18.5 - 3.95i)T^{2}
23 1+(7.99−0.419i)T+(22.8−2.40i)T2 1 + (7.99 - 0.419i)T + (22.8 - 2.40i)T^{2}
29 1+(−0.927−1.27i)T+(−8.96+27.5i)T2 1 + (-0.927 - 1.27i)T + (-8.96 + 27.5i)T^{2}
31 1+(3.58+8.05i)T+(−20.7+23.0i)T2 1 + (3.58 + 8.05i)T + (-20.7 + 23.0i)T^{2}
37 1+(1.62−2.50i)T+(−15.0−33.8i)T2 1 + (1.62 - 2.50i)T + (-15.0 - 33.8i)T^{2}
41 1+(−11.6−3.78i)T+(33.1+24.0i)T2 1 + (-11.6 - 3.78i)T + (33.1 + 24.0i)T^{2}
43 1+(4.19+4.19i)T+43iT2 1 + (4.19 + 4.19i)T + 43iT^{2}
47 1+(2.18+5.68i)T+(−34.9+31.4i)T2 1 + (2.18 + 5.68i)T + (-34.9 + 31.4i)T^{2}
53 1+(2.51−3.10i)T+(−11.0−51.8i)T2 1 + (2.51 - 3.10i)T + (-11.0 - 51.8i)T^{2}
59 1+(2.98+3.31i)T+(−6.16+58.6i)T2 1 + (2.98 + 3.31i)T + (-6.16 + 58.6i)T^{2}
61 1+(−6.76−6.08i)T+(6.37+60.6i)T2 1 + (-6.76 - 6.08i)T + (6.37 + 60.6i)T^{2}
67 1+(−1.57+4.11i)T+(−49.7−44.8i)T2 1 + (-1.57 + 4.11i)T + (-49.7 - 44.8i)T^{2}
71 1+(−6.77+4.91i)T+(21.9−67.5i)T2 1 + (-6.77 + 4.91i)T + (21.9 - 67.5i)T^{2}
73 1+(3.04−1.97i)T+(29.6−66.6i)T2 1 + (3.04 - 1.97i)T + (29.6 - 66.6i)T^{2}
79 1+(1.50−3.38i)T+(−52.8−58.7i)T2 1 + (1.50 - 3.38i)T + (-52.8 - 58.7i)T^{2}
83 1+(−0.0807+0.0127i)T+(78.9−25.6i)T2 1 + (-0.0807 + 0.0127i)T + (78.9 - 25.6i)T^{2}
89 1+(−10.7+11.9i)T+(−9.30−88.5i)T2 1 + (-10.7 + 11.9i)T + (-9.30 - 88.5i)T^{2}
97 1+(−5.33−0.844i)T+(92.2+29.9i)T2 1 + (-5.33 - 0.844i)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

−12.50858218549619572768439201253, −11.67136484487784962927691246068, −10.22882409664185781641857225674, −9.670866678859924264192571862619, −8.993647666307694606127455277454, −7.87060022859823523963572984856, −6.07865377969781415250386143307, −4.10988781139594189266387563211, −3.59856069422125932089156893034, −2.03410171525777927263358635214, 2.45476617106814728836594947615, 3.44577640357779218293343461334, 6.03669360331535512403184403425, 6.80429568534588143541461286925, 7.50781966016923249028267675469, 8.358100944320597604258083286509, 9.472515163133284369208131138043, 10.81761015851968560428832156913, 12.12888035413503956771202331496, 13.14388624911349000937303895287

Graph of the ZZ-function along the critical line