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Properties

Label 2-175-175.108-c1-0-5
Degree 22
Conductor 175175
Sign 0.860−0.509i0.860 - 0.509i
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  + (0.000898 + 0.0171i)2-s + (0.879 + 0.711i)3-s + (1.98 − 0.209i)4-s + (1.43 + 1.71i)5-s + (−0.0114 + 0.0157i)6-s + (−2.51 + 0.822i)7-s + (0.0107 + 0.0678i)8-s + (−0.357 − 1.68i)9-s + (−0.0281 + 0.0261i)10-s + (−4.96 − 1.05i)11-s + (1.89 + 1.23i)12-s + (0.145 + 0.286i)13-s + (−0.0163 − 0.0423i)14-s + (0.0372 + 2.52i)15-s + (3.91 − 0.831i)16-s + (3.48 + 1.33i)17-s + ⋯
L(s)  = 1  + (0.000635 + 0.0121i)2-s + (0.507 + 0.410i)3-s + (0.994 − 0.104i)4-s + (0.640 + 0.767i)5-s + (−0.00466 + 0.00641i)6-s + (−0.950 + 0.310i)7-s + (0.00379 + 0.0239i)8-s + (−0.119 − 0.561i)9-s + (−0.00890 + 0.00825i)10-s + (−1.49 − 0.317i)11-s + (0.547 + 0.355i)12-s + (0.0404 + 0.0793i)13-s + (−0.00437 − 0.0113i)14-s + (0.00961 + 0.652i)15-s + (0.977 − 0.207i)16-s + (0.846 + 0.324i)17-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=((0.860−0.509i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+1/2) L(s)=((0.860−0.509i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 0.860−0.509i0.860 - 0.509i
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), 0.860−0.509i)(2,\ 175,\ (\ :1/2),\ 0.860 - 0.509i)

Particular Values

L(1)L(1) ≈\approx 1.49199+0.408863i1.49199 + 0.408863i
L(12)L(\frac12) ≈\approx 1.49199+0.408863i1.49199 + 0.408863i
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+(−1.43−1.71i)T 1 + (-1.43 - 1.71i)T
7 1+(2.51−0.822i)T 1 + (2.51 - 0.822i)T
good2 1+(−0.000898−0.0171i)T+(−1.98+0.209i)T2 1 + (-0.000898 - 0.0171i)T + (-1.98 + 0.209i)T^{2}
3 1+(−0.879−0.711i)T+(0.623+2.93i)T2 1 + (-0.879 - 0.711i)T + (0.623 + 2.93i)T^{2}
11 1+(4.96+1.05i)T+(10.0+4.47i)T2 1 + (4.96 + 1.05i)T + (10.0 + 4.47i)T^{2}
13 1+(−0.145−0.286i)T+(−7.64+10.5i)T2 1 + (-0.145 - 0.286i)T + (-7.64 + 10.5i)T^{2}
17 1+(−3.48−1.33i)T+(12.6+11.3i)T2 1 + (-3.48 - 1.33i)T + (12.6 + 11.3i)T^{2}
19 1+(−0.698+6.64i)T+(−18.5−3.95i)T2 1 + (-0.698 + 6.64i)T + (-18.5 - 3.95i)T^{2}
23 1+(3.41−0.178i)T+(22.8−2.40i)T2 1 + (3.41 - 0.178i)T + (22.8 - 2.40i)T^{2}
29 1+(−1.99−2.74i)T+(−8.96+27.5i)T2 1 + (-1.99 - 2.74i)T + (-8.96 + 27.5i)T^{2}
31 1+(−0.305−0.686i)T+(−20.7+23.0i)T2 1 + (-0.305 - 0.686i)T + (-20.7 + 23.0i)T^{2}
37 1+(−1.79+2.76i)T+(−15.0−33.8i)T2 1 + (-1.79 + 2.76i)T + (-15.0 - 33.8i)T^{2}
41 1+(7.88+2.56i)T+(33.1+24.0i)T2 1 + (7.88 + 2.56i)T + (33.1 + 24.0i)T^{2}
43 1+(3.99+3.99i)T+43iT2 1 + (3.99 + 3.99i)T + 43iT^{2}
47 1+(−2.25−5.87i)T+(−34.9+31.4i)T2 1 + (-2.25 - 5.87i)T + (-34.9 + 31.4i)T^{2}
53 1+(7.10−8.77i)T+(−11.0−51.8i)T2 1 + (7.10 - 8.77i)T + (-11.0 - 51.8i)T^{2}
59 1+(−6.39−7.10i)T+(−6.16+58.6i)T2 1 + (-6.39 - 7.10i)T + (-6.16 + 58.6i)T^{2}
61 1+(2.03+1.83i)T+(6.37+60.6i)T2 1 + (2.03 + 1.83i)T + (6.37 + 60.6i)T^{2}
67 1+(2.33−6.09i)T+(−49.7−44.8i)T2 1 + (2.33 - 6.09i)T + (-49.7 - 44.8i)T^{2}
71 1+(3.66−2.66i)T+(21.9−67.5i)T2 1 + (3.66 - 2.66i)T + (21.9 - 67.5i)T^{2}
73 1+(8.47−5.50i)T+(29.6−66.6i)T2 1 + (8.47 - 5.50i)T + (29.6 - 66.6i)T^{2}
79 1+(−3.33+7.49i)T+(−52.8−58.7i)T2 1 + (-3.33 + 7.49i)T + (-52.8 - 58.7i)T^{2}
83 1+(−1.28+0.203i)T+(78.9−25.6i)T2 1 + (-1.28 + 0.203i)T + (78.9 - 25.6i)T^{2}
89 1+(0.663−0.736i)T+(−9.30−88.5i)T2 1 + (0.663 - 0.736i)T + (-9.30 - 88.5i)T^{2}
97 1+(−12.2−1.94i)T+(92.2+29.9i)T2 1 + (-12.2 - 1.94i)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

−12.84426385777458488668130478554, −11.74086118890894284955963938436, −10.51934637594232292451721761370, −10.04934079474519669448816629675, −8.907294465428069008298665757443, −7.47830788343433934619275486625, −6.44671199583088401110821930309, −5.56493415795121676656620581182, −3.23969300787659928651361902027, −2.61475271533266398075203215745, 1.93646819375693468704640729691, 3.16147134406834706492569209936, 5.25020036691425434098906773826, 6.29735426625348077707173435354, 7.67046343440579764928264903865, 8.202982629885491898957629420742, 9.989942233781118261312388872584, 10.26504654039952296917924581341, 11.89911602361444792361146723901, 12.78721594799821032663945749194

Graph of the ZZ-function along the critical line