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Properties

Label 2-735-49.36-c1-0-21
Degree 22
Conductor 735735
Sign 0.999+0.0165i0.999 + 0.0165i
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.105 − 0.460i)2-s + (0.623 + 0.781i)3-s + (1.60 − 0.771i)4-s + (0.623 + 0.781i)5-s + (0.294 − 0.369i)6-s + (−2.25 − 1.39i)7-s + (−1.11 − 1.39i)8-s + (−0.222 + 0.974i)9-s + (0.294 − 0.369i)10-s + (1.04 + 4.56i)11-s + (1.60 + 0.771i)12-s + (0.142 + 0.623i)13-s + (−0.403 + 1.18i)14-s + (−0.222 + 0.974i)15-s + (1.69 − 2.12i)16-s + (2.90 + 1.39i)17-s + ⋯
L(s)  = 1  + (−0.0742 − 0.325i)2-s + (0.359 + 0.451i)3-s + (0.800 − 0.385i)4-s + (0.278 + 0.349i)5-s + (0.120 − 0.150i)6-s + (−0.850 − 0.525i)7-s + (−0.393 − 0.492i)8-s + (−0.0741 + 0.324i)9-s + (0.0930 − 0.116i)10-s + (0.313 + 1.37i)11-s + (0.462 + 0.222i)12-s + (0.0394 + 0.172i)13-s + (−0.107 + 0.315i)14-s + (−0.0574 + 0.251i)15-s + (0.422 − 0.530i)16-s + (0.704 + 0.339i)17-s + ⋯

Functional equation

Λ(s)=(735s/2 ΓC(s) L(s)=((0.999+0.0165i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(735s/2 ΓC(s+1/2) L(s)=((0.999+0.0165i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 735735    =    3⋅5⋅723 \cdot 5 \cdot 7^{2}
Sign: 0.999+0.0165i0.999 + 0.0165i
Analytic conductor: 5.869005.86900
Root analytic conductor: 2.422602.42260
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ735(526,⋅)\chi_{735} (526, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 735, ( :1/2), 0.999+0.0165i)(2,\ 735,\ (\ :1/2),\ 0.999 + 0.0165i)

Particular Values

L(1)L(1) ≈\approx 2.03432−0.0168692i2.03432 - 0.0168692i
L(12)L(\frac12) ≈\approx 2.03432−0.0168692i2.03432 - 0.0168692i
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+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
5 1+(−0.623−0.781i)T 1 + (-0.623 - 0.781i)T
7 1+(2.25+1.39i)T 1 + (2.25 + 1.39i)T
good2 1+(0.105+0.460i)T+(−1.80+0.867i)T2 1 + (0.105 + 0.460i)T + (-1.80 + 0.867i)T^{2}
11 1+(−1.04−4.56i)T+(−9.91+4.77i)T2 1 + (-1.04 - 4.56i)T + (-9.91 + 4.77i)T^{2}
13 1+(−0.142−0.623i)T+(−11.7+5.64i)T2 1 + (-0.142 - 0.623i)T + (-11.7 + 5.64i)T^{2}
17 1+(−2.90−1.39i)T+(10.5+13.2i)T2 1 + (-2.90 - 1.39i)T + (10.5 + 13.2i)T^{2}
19 1−8.07T+19T2 1 - 8.07T + 19T^{2}
23 1+(−7.66+3.69i)T+(14.3−17.9i)T2 1 + (-7.66 + 3.69i)T + (14.3 - 17.9i)T^{2}
29 1+(−0.216−0.104i)T+(18.0+22.6i)T2 1 + (-0.216 - 0.104i)T + (18.0 + 22.6i)T^{2}
31 1+3.02T+31T2 1 + 3.02T + 31T^{2}
37 1+(7.07+3.40i)T+(23.0+28.9i)T2 1 + (7.07 + 3.40i)T + (23.0 + 28.9i)T^{2}
41 1+(−3.23−4.05i)T+(−9.12+39.9i)T2 1 + (-3.23 - 4.05i)T + (-9.12 + 39.9i)T^{2}
43 1+(−0.0362+0.0454i)T+(−9.56−41.9i)T2 1 + (-0.0362 + 0.0454i)T + (-9.56 - 41.9i)T^{2}
47 1+(0.410+1.80i)T+(−42.3+20.3i)T2 1 + (0.410 + 1.80i)T + (-42.3 + 20.3i)T^{2}
53 1+(−1.50+0.726i)T+(33.0−41.4i)T2 1 + (-1.50 + 0.726i)T + (33.0 - 41.4i)T^{2}
59 1+(−1.86+2.33i)T+(−13.1−57.5i)T2 1 + (-1.86 + 2.33i)T + (-13.1 - 57.5i)T^{2}
61 1+(8.53+4.10i)T+(38.0+47.6i)T2 1 + (8.53 + 4.10i)T + (38.0 + 47.6i)T^{2}
67 1+15.6T+67T2 1 + 15.6T + 67T^{2}
71 1+(10.7−5.17i)T+(44.2−55.5i)T2 1 + (10.7 - 5.17i)T + (44.2 - 55.5i)T^{2}
73 1+(−2.95+12.9i)T+(−65.7−31.6i)T2 1 + (-2.95 + 12.9i)T + (-65.7 - 31.6i)T^{2}
79 1+10.1T+79T2 1 + 10.1T + 79T^{2}
83 1+(3.07−13.4i)T+(−74.7−36.0i)T2 1 + (3.07 - 13.4i)T + (-74.7 - 36.0i)T^{2}
89 1+(0.0120−0.0527i)T+(−80.1−38.6i)T2 1 + (0.0120 - 0.0527i)T + (-80.1 - 38.6i)T^{2}
97 1−10.9T+97T2 1 - 10.9T + 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

−10.27912766131637605465676692091, −9.703042718266920338237986438940, −9.114954939997587591343610968107, −7.37141803570647728265220295477, −7.11806300450954308909082098797, −6.02196776499603777513865638469, −4.90905872601791633134579371782, −3.54404529236496904385407762750, −2.80340614861400570806186910927, −1.42480787025848217506093687587, 1.26676568925972573799009921016, 3.04071083099275277938334365111, 3.24861409540024308061056999059, 5.46709809940542005442166158074, 5.91828518921764037126155783369, 7.04013028852719495342181667291, 7.61500775815346368318068187644, 8.796952972634559540663982951381, 9.148097324942187794529253721707, 10.33063418397588565269610397713

Graph of the ZZ-function along the critical line