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Properties

Label 2-435-145.102-c1-0-4
Degree 22
Conductor 435435
Sign 0.996+0.0868i0.996 + 0.0868i
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  + (−1.01 − 0.806i)2-s + (−0.222 + 0.974i)3-s + (−0.0726 − 0.318i)4-s + (−1.73 − 1.41i)5-s + (1.01 − 0.806i)6-s + (−2.96 + 1.86i)7-s + (−1.30 + 2.71i)8-s + (−0.900 − 0.433i)9-s + (0.610 + 2.82i)10-s + (4.30 + 1.50i)11-s + 0.326·12-s + (2.94 + 1.03i)13-s + (4.49 + 0.506i)14-s + (1.76 − 1.37i)15-s + (2.91 − 1.40i)16-s − 7.34i·17-s + ⋯
L(s)  = 1  + (−0.715 − 0.570i)2-s + (−0.128 + 0.562i)3-s + (−0.0363 − 0.159i)4-s + (−0.774 − 0.632i)5-s + (0.412 − 0.329i)6-s + (−1.12 + 0.703i)7-s + (−0.461 + 0.958i)8-s + (−0.300 − 0.144i)9-s + (0.193 + 0.894i)10-s + (1.29 + 0.454i)11-s + 0.0942·12-s + (0.818 + 0.286i)13-s + (1.20 + 0.135i)14-s + (0.455 − 0.354i)15-s + (0.729 − 0.351i)16-s − 1.78i·17-s + ⋯

Functional equation

Λ(s)=(435s/2 ΓC(s) L(s)=((0.996+0.0868i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0868i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(435s/2 ΓC(s+1/2) L(s)=((0.996+0.0868i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 435435    =    3⋅5⋅293 \cdot 5 \cdot 29
Sign: 0.996+0.0868i0.996 + 0.0868i
Analytic conductor: 3.473493.47349
Root analytic conductor: 1.863731.86373
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ435(247,⋅)\chi_{435} (247, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 435, ( :1/2), 0.996+0.0868i)(2,\ 435,\ (\ :1/2),\ 0.996 + 0.0868i)

Particular Values

L(1)L(1) ≈\approx 0.682394−0.0296809i0.682394 - 0.0296809i
L(12)L(\frac12) ≈\approx 0.682394−0.0296809i0.682394 - 0.0296809i
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+(1.73+1.41i)T 1 + (1.73 + 1.41i)T
29 1+(−3.56+4.03i)T 1 + (-3.56 + 4.03i)T
good2 1+(1.01+0.806i)T+(0.445+1.94i)T2 1 + (1.01 + 0.806i)T + (0.445 + 1.94i)T^{2}
7 1+(2.96−1.86i)T+(3.03−6.30i)T2 1 + (2.96 - 1.86i)T + (3.03 - 6.30i)T^{2}
11 1+(−4.30−1.50i)T+(8.60+6.85i)T2 1 + (-4.30 - 1.50i)T + (8.60 + 6.85i)T^{2}
13 1+(−2.94−1.03i)T+(10.1+8.10i)T2 1 + (-2.94 - 1.03i)T + (10.1 + 8.10i)T^{2}
17 1+7.34iT−17T2 1 + 7.34iT - 17T^{2}
19 1+(−7.34−4.61i)T+(8.24+17.1i)T2 1 + (-7.34 - 4.61i)T + (8.24 + 17.1i)T^{2}
23 1+(2.83+0.319i)T+(22.4+5.11i)T2 1 + (2.83 + 0.319i)T + (22.4 + 5.11i)T^{2}
31 1+(2.56−0.289i)T+(30.2−6.89i)T2 1 + (2.56 - 0.289i)T + (30.2 - 6.89i)T^{2}
37 1+(−2.16−1.04i)T+(23.0+28.9i)T2 1 + (-2.16 - 1.04i)T + (23.0 + 28.9i)T^{2}
41 1+(−1.02+1.02i)T−41iT2 1 + (-1.02 + 1.02i)T - 41iT^{2}
43 1+(−6.71−8.41i)T+(−9.56+41.9i)T2 1 + (-6.71 - 8.41i)T + (-9.56 + 41.9i)T^{2}
47 1+(1.48−0.717i)T+(29.3−36.7i)T2 1 + (1.48 - 0.717i)T + (29.3 - 36.7i)T^{2}
53 1+(−1.03−9.15i)T+(−51.6+11.7i)T2 1 + (-1.03 - 9.15i)T + (-51.6 + 11.7i)T^{2}
59 1−5.26iT−59T2 1 - 5.26iT - 59T^{2}
61 1+(−2.95+1.85i)T+(26.4−54.9i)T2 1 + (-2.95 + 1.85i)T + (26.4 - 54.9i)T^{2}
67 1+(−1.61+0.565i)T+(52.3−41.7i)T2 1 + (-1.61 + 0.565i)T + (52.3 - 41.7i)T^{2}
71 1+(−2.50−5.21i)T+(−44.2+55.5i)T2 1 + (-2.50 - 5.21i)T + (-44.2 + 55.5i)T^{2}
73 1+(−2.64+2.10i)T+(16.2−71.1i)T2 1 + (-2.64 + 2.10i)T + (16.2 - 71.1i)T^{2}
79 1+(6.21−2.17i)T+(61.7−49.2i)T2 1 + (6.21 - 2.17i)T + (61.7 - 49.2i)T^{2}
83 1+(−12.9−8.11i)T+(36.0+74.7i)T2 1 + (-12.9 - 8.11i)T + (36.0 + 74.7i)T^{2}
89 1+(0.663+5.89i)T+(−86.7+19.8i)T2 1 + (0.663 + 5.89i)T + (-86.7 + 19.8i)T^{2}
97 1+(−2.29−10.0i)T+(−87.3+42.0i)T2 1 + (-2.29 - 10.0i)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

−11.26311310401370525131349471182, −9.851137539080215937851616648010, −9.422231739551016733627316227097, −8.938579593721907588304895003835, −7.72171718714232037563347740023, −6.31392287607449597667598589308, −5.37115083463936301507680344055, −4.11187790813832812710876022426, −2.96049433477573023494131066366, −1.03420909586196436672030533221, 0.77307159473631679681489060558, 3.40684364144982076730562836904, 3.78347023144170653454353395347, 6.10826952760700630615326915634, 6.70340241087846672369078559162, 7.37430125586133399112942322919, 8.306775992300551551281548113156, 9.113730125597579438596352471401, 10.16856198953398775563290604659, 11.11696041449492555936107015452

Graph of the ZZ-function along the critical line