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Properties

Label 2-735-21.17-c1-0-48
Degree 22
Conductor 735735
Sign −0.683−0.729i-0.683 - 0.729i
Analytic cond. 5.869005.86900
Root an. cond. 2.422602.42260
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.686 − 0.396i)2-s + (−0.5 − 1.65i)3-s + (−0.686 + 1.18i)4-s + (−0.5 − 0.866i)5-s + (−1 − 0.939i)6-s + 2.67i·8-s + (−2.5 + 1.65i)9-s + (−0.686 − 0.396i)10-s + (−2.18 − 1.26i)11-s + (2.31 + 0.543i)12-s − 4.10i·13-s + (−1.18 + 1.26i)15-s + (−0.313 − 0.543i)16-s + (−2.18 + 3.78i)17-s + (−1.05 + 2.12i)18-s + (−3 + 1.73i)19-s + ⋯
L(s)  = 1  + (0.485 − 0.280i)2-s + (−0.288 − 0.957i)3-s + (−0.343 + 0.594i)4-s + (−0.223 − 0.387i)5-s + (−0.408 − 0.383i)6-s + 0.944i·8-s + (−0.833 + 0.552i)9-s + (−0.216 − 0.125i)10-s + (−0.659 − 0.380i)11-s + (0.667 + 0.156i)12-s − 1.13i·13-s + (−0.306 + 0.325i)15-s + (−0.0784 − 0.135i)16-s + (−0.530 + 0.918i)17-s + (−0.249 + 0.501i)18-s + (−0.688 + 0.397i)19-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((−0.683−0.729i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((−0.683−0.729i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: −0.683−0.729i-0.683 - 0.729i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(521,⋅)\chi_{735} (521, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 735, ( :1/2), −0.683−0.729i)(2,\ 735,\ (\ :1/2),\ -0.683 - 0.729i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+(0.5+1.65i)T 1 + (0.5 + 1.65i)T
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1 1
good2 1+(−0.686+0.396i)T+(1−1.73i)T2 1 + (-0.686 + 0.396i)T + (1 - 1.73i)T^{2}
11 1+(2.18+1.26i)T+(5.5+9.52i)T2 1 + (2.18 + 1.26i)T + (5.5 + 9.52i)T^{2}
13 1+4.10iT−13T2 1 + 4.10iT - 13T^{2}
17 1+(2.18−3.78i)T+(−8.5−14.7i)T2 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2}
19 1+(3−1.73i)T+(9.5−16.4i)T2 1 + (3 - 1.73i)T + (9.5 - 16.4i)T^{2}
23 1+(7.37−4.25i)T+(11.5−19.9i)T2 1 + (7.37 - 4.25i)T + (11.5 - 19.9i)T^{2}
29 1+0.939iT−29T2 1 + 0.939iT - 29T^{2}
31 1+(3+1.73i)T+(15.5+26.8i)T2 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2}
37 1+(−3.37−5.84i)T+(−18.5+32.0i)T2 1 + (-3.37 - 5.84i)T + (-18.5 + 32.0i)T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4.74T+43T2 1 - 4.74T + 43T^{2}
47 1+(−0.813−1.40i)T+(−23.5+40.7i)T2 1 + (-0.813 - 1.40i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.62+0.939i)T+(26.5+45.8i)T2 1 + (1.62 + 0.939i)T + (26.5 + 45.8i)T^{2}
59 1+(−4.37+7.57i)T+(−29.5−51.0i)T2 1 + (-4.37 + 7.57i)T + (-29.5 - 51.0i)T^{2}
61 1+(−6+3.46i)T+(30.5−52.8i)T2 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2}
67 1+(2.37−4.10i)T+(−33.5−58.0i)T2 1 + (2.37 - 4.10i)T + (-33.5 - 58.0i)T^{2}
71 1+0.294iT−71T2 1 + 0.294iT - 71T^{2}
73 1+(6+3.46i)T+(36.5+63.2i)T2 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2}
79 1+(−1.18−2.05i)T+(−39.5+68.4i)T2 1 + (-1.18 - 2.05i)T + (-39.5 + 68.4i)T^{2}
83 1+17.4T+83T2 1 + 17.4T + 83T^{2}
89 1+(7.37+12.7i)T+(−44.5+77.0i)T2 1 + (7.37 + 12.7i)T + (-44.5 + 77.0i)T^{2}
97 1+11.0iT−97T2 1 + 11.0iT - 97T^{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

−9.993520351858491363006438214363, −8.443609824283572369113302410814, −8.238031433223495609439805565836, −7.43605306084102291681294566740, −6.03882635594684108175454790410, −5.42473902684477284679254674194, −4.23488849194746780979303160692, −3.14665007656182472523434853285, −1.94704036782126069818120258769, 0, 2.42043711468817377514870868992, 4.00077382364220416236385425962, 4.49292449248453140819467458082, 5.43987791663344200418010095689, 6.36647405910844552689341622122, 7.14573780348509903700571580083, 8.593490888608196655393025055681, 9.351901992941828904131818680931, 10.12919248947150458968448577343

Graph of the ZZ-function along the critical line