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Properties

Label 2-700-35.19-c4-0-44
Degree 22
Conductor 700700
Sign −0.999−0.0166i-0.999 - 0.0166i
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  + (0.252 + 0.437i)3-s + (48.2 − 8.29i)7-s + (40.3 − 69.9i)9-s + (−106. − 183. i)11-s − 45.2·13-s + (125. + 217. i)17-s + (−405. − 234. i)19-s + (15.8 + 19.0i)21-s + (−809. − 467. i)23-s + 81.7·27-s + 816.·29-s + (−853. + 492. i)31-s + (53.6 − 93.0i)33-s + (−474. − 273. i)37-s + (−11.4 − 19.8i)39-s + ⋯
L(s)  = 1  + (0.0280 + 0.0486i)3-s + (0.985 − 0.169i)7-s + (0.498 − 0.863i)9-s + (−0.877 − 1.51i)11-s − 0.267·13-s + (0.435 + 0.753i)17-s + (−1.12 − 0.649i)19-s + (0.0359 + 0.0432i)21-s + (−1.52 − 0.883i)23-s + 0.112·27-s + 0.970·29-s + (−0.888 + 0.512i)31-s + (0.0493 − 0.0854i)33-s + (−0.346 − 0.200i)37-s + (−0.00751 − 0.0130i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.999−0.0166i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0166i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.999−0.0166i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0166i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.999−0.0166i-0.999 - 0.0166i
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), −0.999−0.0166i)(2,\ 700,\ (\ :2),\ -0.999 - 0.0166i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 0.65397490420.6539749042
L(12)L(\frac12) ≈\approx 0.65397490420.6539749042
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+(−48.2+8.29i)T 1 + (-48.2 + 8.29i)T
good3 1+(−0.252−0.437i)T+(−40.5+70.1i)T2 1 + (-0.252 - 0.437i)T + (-40.5 + 70.1i)T^{2}
11 1+(106.+183.i)T+(−7.32e3+1.26e4i)T2 1 + (106. + 183. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1+45.2T+2.85e4T2 1 + 45.2T + 2.85e4T^{2}
17 1+(−125.−217.i)T+(−4.17e4+7.23e4i)T2 1 + (-125. - 217. i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(405.+234.i)T+(6.51e4+1.12e5i)T2 1 + (405. + 234. i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(809.+467.i)T+(1.39e5+2.42e5i)T2 1 + (809. + 467. i)T + (1.39e5 + 2.42e5i)T^{2}
29 1−816.T+7.07e5T2 1 - 816.T + 7.07e5T^{2}
31 1+(853.−492.i)T+(4.61e5−7.99e5i)T2 1 + (853. - 492. i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(474.+273.i)T+(9.37e5+1.62e6i)T2 1 + (474. + 273. i)T + (9.37e5 + 1.62e6i)T^{2}
41 1−2.48e3iT−2.82e6T2 1 - 2.48e3iT - 2.82e6T^{2}
43 1−1.88e3iT−3.41e6T2 1 - 1.88e3iT - 3.41e6T^{2}
47 1+(289.−502.i)T+(−2.43e6−4.22e6i)T2 1 + (289. - 502. i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(−312.+180.i)T+(3.94e6−6.83e6i)T2 1 + (-312. + 180. i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(5.28e3−3.04e3i)T+(6.05e6−1.04e7i)T2 1 + (5.28e3 - 3.04e3i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(1.53e3+884.i)T+(6.92e6+1.19e7i)T2 1 + (1.53e3 + 884. i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(−5.17e3+2.98e3i)T+(1.00e7−1.74e7i)T2 1 + (-5.17e3 + 2.98e3i)T + (1.00e7 - 1.74e7i)T^{2}
71 1−1.01e3T+2.54e7T2 1 - 1.01e3T + 2.54e7T^{2}
73 1+(−1.54e3−2.67e3i)T+(−1.41e7+2.45e7i)T2 1 + (-1.54e3 - 2.67e3i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(361.−626.i)T+(−1.94e7−3.37e7i)T2 1 + (361. - 626. i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1−8.10e3T+4.74e7T2 1 - 8.10e3T + 4.74e7T^{2}
89 1+(2.39e3+1.37e3i)T+(3.13e7+5.43e7i)T2 1 + (2.39e3 + 1.37e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1+1.66e4T+8.85e7T2 1 + 1.66e4T + 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

−9.404793677081057377230471096318, −8.269941805822597456104022390185, −8.073338295248116279648256774707, −6.65381132398494750412805973188, −5.91168364567639511049584130551, −4.78078565098738053658186782146, −3.87762743283480979419319982791, −2.68922522822876145095283251505, −1.33623318424690748230803426882, −0.14517623425416147251304888355, 1.79316072047752613109972565164, 2.25942138427974275606151525352, 4.05460027612460233674284742648, 4.88570367478690676024550198386, 5.58524394495989462102404095720, 7.08950698458636393712207650342, 7.70329111394387442768828494854, 8.324003014229644731283010638073, 9.636504795890850777885651414445, 10.28561541615328140769309047241

Graph of the ZZ-function along the critical line