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Properties

Label 2-700-35.24-c4-0-39
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(549,⋅)\chi_{700} (549, \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

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

Graph of the ZZ-function along the critical line