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Properties

Label 2-435-29.24-c1-0-18
Degree 22
Conductor 435435
Sign −0.385+0.922i-0.385 + 0.922i
Analytic cond. 3.473493.47349
Root an. cond. 1.863731.86373
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  + (0.320 + 0.401i)2-s + (−0.222 − 0.974i)3-s + (0.386 − 1.69i)4-s + (−0.623 − 0.781i)5-s + (0.320 − 0.401i)6-s + (−0.169 − 0.744i)7-s + (1.72 − 0.832i)8-s + (−0.900 + 0.433i)9-s + (0.114 − 0.500i)10-s + (−0.761 − 0.366i)11-s − 1.73·12-s + (−5.11 − 2.46i)13-s + (0.244 − 0.306i)14-s + (−0.623 + 0.781i)15-s + (−2.24 − 1.07i)16-s + 5.13·17-s + ⋯
L(s)  = 1  + (0.226 + 0.283i)2-s + (−0.128 − 0.562i)3-s + (0.193 − 0.846i)4-s + (−0.278 − 0.349i)5-s + (0.130 − 0.163i)6-s + (−0.0641 − 0.281i)7-s + (0.611 − 0.294i)8-s + (−0.300 + 0.144i)9-s + (0.0361 − 0.158i)10-s + (−0.229 − 0.110i)11-s − 0.501·12-s + (−1.41 − 0.682i)13-s + (0.0653 − 0.0818i)14-s + (−0.160 + 0.201i)15-s + (−0.560 − 0.269i)16-s + 1.24·17-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((−0.385+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((−0.385+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: −0.385+0.922i-0.385 + 0.922i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(256,⋅)\chi_{435} (256, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), −0.385+0.922i)(2,\ 435,\ (\ :1/2),\ -0.385 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 0.697113−1.04673i0.697113 - 1.04673i
L(12)L(\frac12) ≈\approx 0.697113−1.04673i0.697113 - 1.04673i
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)
bad3 1+(0.222+0.974i)T 1 + (0.222 + 0.974i)T
5 1+(0.623+0.781i)T 1 + (0.623 + 0.781i)T
29 1+(4.76+2.50i)T 1 + (4.76 + 2.50i)T
good2 1+(−0.320−0.401i)T+(−0.445+1.94i)T2 1 + (-0.320 - 0.401i)T + (-0.445 + 1.94i)T^{2}
7 1+(0.169+0.744i)T+(−6.30+3.03i)T2 1 + (0.169 + 0.744i)T + (-6.30 + 3.03i)T^{2}
11 1+(0.761+0.366i)T+(6.85+8.60i)T2 1 + (0.761 + 0.366i)T + (6.85 + 8.60i)T^{2}
13 1+(5.11+2.46i)T+(8.10+10.1i)T2 1 + (5.11 + 2.46i)T + (8.10 + 10.1i)T^{2}
17 1−5.13T+17T2 1 - 5.13T + 17T^{2}
19 1+(1.17−5.14i)T+(−17.1−8.24i)T2 1 + (1.17 - 5.14i)T + (-17.1 - 8.24i)T^{2}
23 1+(−2.92+3.67i)T+(−5.11−22.4i)T2 1 + (-2.92 + 3.67i)T + (-5.11 - 22.4i)T^{2}
31 1+(4.36+5.47i)T+(−6.89+30.2i)T2 1 + (4.36 + 5.47i)T + (-6.89 + 30.2i)T^{2}
37 1+(1.28−0.619i)T+(23.0−28.9i)T2 1 + (1.28 - 0.619i)T + (23.0 - 28.9i)T^{2}
41 1−6.49T+41T2 1 - 6.49T + 41T^{2}
43 1+(−6.89+8.65i)T+(−9.56−41.9i)T2 1 + (-6.89 + 8.65i)T + (-9.56 - 41.9i)T^{2}
47 1+(−11.3−5.47i)T+(29.3+36.7i)T2 1 + (-11.3 - 5.47i)T + (29.3 + 36.7i)T^{2}
53 1+(−1.53−1.92i)T+(−11.7+51.6i)T2 1 + (-1.53 - 1.92i)T + (-11.7 + 51.6i)T^{2}
59 1−2.12T+59T2 1 - 2.12T + 59T^{2}
61 1+(−2.91−12.7i)T+(−54.9+26.4i)T2 1 + (-2.91 - 12.7i)T + (-54.9 + 26.4i)T^{2}
67 1+(3.74−1.80i)T+(41.7−52.3i)T2 1 + (3.74 - 1.80i)T + (41.7 - 52.3i)T^{2}
71 1+(3.60+1.73i)T+(44.2+55.5i)T2 1 + (3.60 + 1.73i)T + (44.2 + 55.5i)T^{2}
73 1+(−6.28+7.88i)T+(−16.2−71.1i)T2 1 + (-6.28 + 7.88i)T + (-16.2 - 71.1i)T^{2}
79 1+(10.6−5.12i)T+(49.2−61.7i)T2 1 + (10.6 - 5.12i)T + (49.2 - 61.7i)T^{2}
83 1+(−2.12+9.29i)T+(−74.7−36.0i)T2 1 + (-2.12 + 9.29i)T + (-74.7 - 36.0i)T^{2}
89 1+(−6.92−8.68i)T+(−19.8+86.7i)T2 1 + (-6.92 - 8.68i)T + (-19.8 + 86.7i)T^{2}
97 1+(−0.648+2.84i)T+(−87.3−42.0i)T2 1 + (-0.648 + 2.84i)T + (-87.3 - 42.0i)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

−10.70282255422355737702964389837, −10.14905670037249882439781900812, −9.091078110939347197965712297617, −7.59593531199727484542211493553, −7.41302140110402453617442451098, −5.88768014266234273492098076249, −5.40854209761474866728452516745, −4.10908063758663078927997713167, −2.36495539243129858037028279995, −0.74677239802277120144891295224, 2.42379180986097563359225948341, 3.39091626344950183146282798848, 4.50522185737734596632569340892, 5.46171894777573256501759152791, 7.09436134991411338693436941805, 7.52219647319416120396837413297, 8.862592395698007013205669547140, 9.613671981549427577024779389539, 10.77165093601684216222573648151, 11.40171805999219256436945394678

Graph of the ZZ-function along the critical line