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Properties

Label 2-700-28.3-c1-0-47
Degree 22
Conductor 700700
Sign 0.996−0.0821i0.996 - 0.0821i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
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  + (0.501 + 1.32i)2-s + (0.895 − 1.55i)3-s + (−1.49 + 1.32i)4-s + (2.49 + 0.405i)6-s + (0.644 − 2.56i)7-s + (−2.50 − 1.31i)8-s + (−0.103 − 0.179i)9-s + (3.66 + 2.11i)11-s + (0.717 + 3.50i)12-s − 2.98i·13-s + (3.71 − 0.434i)14-s + (0.480 − 3.97i)16-s + (1.92 + 1.10i)17-s + (0.184 − 0.226i)18-s + (−2.28 − 3.95i)19-s + ⋯
L(s)  = 1  + (0.354 + 0.934i)2-s + (0.516 − 0.895i)3-s + (−0.748 + 0.663i)4-s + (1.02 + 0.165i)6-s + (0.243 − 0.969i)7-s + (−0.885 − 0.464i)8-s + (−0.0344 − 0.0596i)9-s + (1.10 + 0.638i)11-s + (0.207 + 1.01i)12-s − 0.827i·13-s + (0.993 − 0.116i)14-s + (0.120 − 0.992i)16-s + (0.465 + 0.268i)17-s + (0.0435 − 0.0533i)18-s + (−0.523 − 0.907i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.996−0.0821i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.996−0.0821i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0821i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.996−0.0821i0.996 - 0.0821i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(451,⋅)\chi_{700} (451, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.996−0.0821i)(2,\ 700,\ (\ :1/2),\ 0.996 - 0.0821i)

Particular Values

L(1)L(1) ≈\approx 2.12288+0.0872949i2.12288 + 0.0872949i
L(12)L(\frac12) ≈\approx 2.12288+0.0872949i2.12288 + 0.0872949i
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)
bad2 1+(−0.501−1.32i)T 1 + (-0.501 - 1.32i)T
5 1 1
7 1+(−0.644+2.56i)T 1 + (-0.644 + 2.56i)T
good3 1+(−0.895+1.55i)T+(−1.5−2.59i)T2 1 + (-0.895 + 1.55i)T + (-1.5 - 2.59i)T^{2}
11 1+(−3.66−2.11i)T+(5.5+9.52i)T2 1 + (-3.66 - 2.11i)T + (5.5 + 9.52i)T^{2}
13 1+2.98iT−13T2 1 + 2.98iT - 13T^{2}
17 1+(−1.92−1.10i)T+(8.5+14.7i)T2 1 + (-1.92 - 1.10i)T + (8.5 + 14.7i)T^{2}
19 1+(2.28+3.95i)T+(−9.5+16.4i)T2 1 + (2.28 + 3.95i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.78+1.02i)T+(11.5−19.9i)T2 1 + (-1.78 + 1.02i)T + (11.5 - 19.9i)T^{2}
29 1−6.42T+29T2 1 - 6.42T + 29T^{2}
31 1+(−1.20+2.07i)T+(−15.5−26.8i)T2 1 + (-1.20 + 2.07i)T + (-15.5 - 26.8i)T^{2}
37 1+(2.16+3.74i)T+(−18.5+32.0i)T2 1 + (2.16 + 3.74i)T + (-18.5 + 32.0i)T^{2}
41 1−4.88iT−41T2 1 - 4.88iT - 41T^{2}
43 1−12.3iT−43T2 1 - 12.3iT - 43T^{2}
47 1+(3.38+5.85i)T+(−23.5+40.7i)T2 1 + (3.38 + 5.85i)T + (-23.5 + 40.7i)T^{2}
53 1+(6.41−11.1i)T+(−26.5−45.8i)T2 1 + (6.41 - 11.1i)T + (-26.5 - 45.8i)T^{2}
59 1+(−6.99+12.1i)T+(−29.5−51.0i)T2 1 + (-6.99 + 12.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(−0.0195+0.0113i)T+(30.5−52.8i)T2 1 + (-0.0195 + 0.0113i)T + (30.5 - 52.8i)T^{2}
67 1+(4.38+2.53i)T+(33.5+58.0i)T2 1 + (4.38 + 2.53i)T + (33.5 + 58.0i)T^{2}
71 1−4.07iT−71T2 1 - 4.07iT - 71T^{2}
73 1+(−2.88−1.66i)T+(36.5+63.2i)T2 1 + (-2.88 - 1.66i)T + (36.5 + 63.2i)T^{2}
79 1+(3.14−1.81i)T+(39.5−68.4i)T2 1 + (3.14 - 1.81i)T + (39.5 - 68.4i)T^{2}
83 1+11.7T+83T2 1 + 11.7T + 83T^{2}
89 1+(14.4−8.34i)T+(44.5−77.0i)T2 1 + (14.4 - 8.34i)T + (44.5 - 77.0i)T^{2}
97 1−12.0iT−97T2 1 - 12.0iT - 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

−10.31257041944142397859058887082, −9.362230510473369215018167816380, −8.287483932398258250211018982704, −7.79838941412977377794160563436, −6.89501959347543161258900324375, −6.44565097158546615312466984459, −4.95117665632687185184084917711, −4.15008657182431615689365663646, −2.88209644780421367917579606380, −1.14505573382865114000488085633, 1.55657833016569898114496574699, 2.93875391056247378422505244744, 3.79055084833694586101740780952, 4.62786508571742246746527241323, 5.68332248785223462161535932020, 6.65827654344685321124084259454, 8.613212008785825140754328955767, 8.743642790260261972329308608428, 9.691259108105234908746986589728, 10.31439175250312382751406875481

Graph of the ZZ-function along the critical line