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Properties

Label 2-700-35.19-c4-0-10
Degree 22
Conductor 700700
Sign 0.0708−0.997i0.0708 - 0.997i
Analytic cond. 72.358972.3589
Root an. cond. 8.506408.50640
Motivic weight 44
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.50 − 7.81i)3-s + (5.65 + 48.6i)7-s + (−0.179 + 0.311i)9-s + (72.8 + 126. i)11-s + 209.·13-s + (93.5 + 162. i)17-s + (−153. − 88.6i)19-s + (354. − 263. i)21-s + (−288. − 166. i)23-s − 727.·27-s + 1.38e3·29-s + (−1.34e3 + 776. i)31-s + (657. − 1.13e3i)33-s + (−1.97e3 − 1.13e3i)37-s + (−946. − 1.63e3i)39-s + ⋯
L(s)  = 1  + (−0.501 − 0.867i)3-s + (0.115 + 0.993i)7-s + (−0.00222 + 0.00384i)9-s + (0.602 + 1.04i)11-s + 1.24·13-s + (0.323 + 0.560i)17-s + (−0.425 − 0.245i)19-s + (0.804 − 0.598i)21-s + (−0.546 − 0.315i)23-s − 0.997·27-s + 1.64·29-s + (−1.40 + 0.808i)31-s + (0.603 − 1.04i)33-s + (−1.44 − 0.831i)37-s + (−0.622 − 1.07i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.0708−0.997i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((0.0708−0.997i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0708 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.0708−0.997i0.0708 - 0.997i
Analytic conductor: 72.358972.3589
Root analytic conductor: 8.506408.50640
Motivic weight: 44
Rational: no
Arithmetic: yes
Character: χ700(649,⋅)\chi_{700} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :2), 0.0708−0.997i)(2,\ 700,\ (\ :2),\ 0.0708 - 0.997i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.2340202121.234020212
L(12)L(\frac12) ≈\approx 1.2340202121.234020212
L(3)L(3) 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 1
5 1 1
7 1+(−5.65−48.6i)T 1 + (-5.65 - 48.6i)T
good3 1+(4.50+7.81i)T+(−40.5+70.1i)T2 1 + (4.50 + 7.81i)T + (-40.5 + 70.1i)T^{2}
11 1+(−72.8−126.i)T+(−7.32e3+1.26e4i)T2 1 + (-72.8 - 126. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1−209.T+2.85e4T2 1 - 209.T + 2.85e4T^{2}
17 1+(−93.5−162.i)T+(−4.17e4+7.23e4i)T2 1 + (-93.5 - 162. i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(153.+88.6i)T+(6.51e4+1.12e5i)T2 1 + (153. + 88.6i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(288.+166.i)T+(1.39e5+2.42e5i)T2 1 + (288. + 166. i)T + (1.39e5 + 2.42e5i)T^{2}
29 1−1.38e3T+7.07e5T2 1 - 1.38e3T + 7.07e5T^{2}
31 1+(1.34e3−776.i)T+(4.61e5−7.99e5i)T2 1 + (1.34e3 - 776. i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(1.97e3+1.13e3i)T+(9.37e5+1.62e6i)T2 1 + (1.97e3 + 1.13e3i)T + (9.37e5 + 1.62e6i)T^{2}
41 1−781.iT−2.82e6T2 1 - 781. iT - 2.82e6T^{2}
43 1−837.iT−3.41e6T2 1 - 837. iT - 3.41e6T^{2}
47 1+(748.−1.29e3i)T+(−2.43e6−4.22e6i)T2 1 + (748. - 1.29e3i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(3.86e3−2.23e3i)T+(3.94e6−6.83e6i)T2 1 + (3.86e3 - 2.23e3i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(−3.81e3+2.20e3i)T+(6.05e6−1.04e7i)T2 1 + (-3.81e3 + 2.20e3i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(−2.25e3−1.30e3i)T+(6.92e6+1.19e7i)T2 1 + (-2.25e3 - 1.30e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(5.00e3−2.88e3i)T+(1.00e7−1.74e7i)T2 1 + (5.00e3 - 2.88e3i)T + (1.00e7 - 1.74e7i)T^{2}
71 1−1.14e3T+2.54e7T2 1 - 1.14e3T + 2.54e7T^{2}
73 1+(1.84e3+3.19e3i)T+(−1.41e7+2.45e7i)T2 1 + (1.84e3 + 3.19e3i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(−1.34e3+2.33e3i)T+(−1.94e7−3.37e7i)T2 1 + (-1.34e3 + 2.33e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1+2.68e3T+4.74e7T2 1 + 2.68e3T + 4.74e7T^{2}
89 1+(5.03e3+2.90e3i)T+(3.13e7+5.43e7i)T2 1 + (5.03e3 + 2.90e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1−7.89e3T+8.85e7T2 1 - 7.89e3T + 8.85e7T^{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

−10.10200106694206550139826647348, −9.042747116551124385140286596027, −8.403717537616932805457122923739, −7.29021158710482696326586654230, −6.42766535422435829932853602677, −5.91006501491857597930851683053, −4.69876977254884902769531021900, −3.50152752871293747137839134766, −1.99725635461651181243845939515, −1.25365166954925303448680959248, 0.33168843631618051188287833601, 1.52509410466711084028451989781, 3.46005352089338137324079979942, 3.99357067200551983541106942316, 5.06824661888265399498281820188, 5.99497405046565798746301666506, 6.89383722989367026059286002470, 8.049742448410723172310598619425, 8.823196446979848072777309958568, 9.920707633848140129150734570677

Graph of the ZZ-function along the critical line