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Properties

Label 2-700-35.19-c4-0-25
Degree 22
Conductor 700700
Sign −0.0137+0.999i-0.0137 + 0.999i
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  + (−2.80 − 4.85i)3-s + (−48.2 + 8.43i)7-s + (24.8 − 42.9i)9-s + (91.4 + 158. i)11-s − 206.·13-s + (−72.5 − 125. i)17-s + (127. + 73.8i)19-s + (176. + 210. i)21-s + (530. + 306. i)23-s − 731.·27-s + 63.1·29-s + (−83.2 + 48.0i)31-s + (512. − 887. i)33-s + (650. + 375. i)37-s + (579. + 1.00e3i)39-s + ⋯
L(s)  = 1  + (−0.311 − 0.539i)3-s + (−0.985 + 0.172i)7-s + (0.306 − 0.530i)9-s + (0.755 + 1.30i)11-s − 1.22·13-s + (−0.250 − 0.434i)17-s + (0.354 + 0.204i)19-s + (0.399 + 0.477i)21-s + (1.00 + 0.578i)23-s − 1.00·27-s + 0.0750·29-s + (−0.0866 + 0.0500i)31-s + (0.470 − 0.814i)33-s + (0.475 + 0.274i)37-s + (0.380 + 0.659i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.0137+0.999i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.0137+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0137 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.0137+0.999i-0.0137 + 0.999i
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), −0.0137+0.999i)(2,\ 700,\ (\ :2),\ -0.0137 + 0.999i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.1553746661.155374666
L(12)L(\frac12) ≈\approx 1.1553746661.155374666
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+(48.2−8.43i)T 1 + (48.2 - 8.43i)T
good3 1+(2.80+4.85i)T+(−40.5+70.1i)T2 1 + (2.80 + 4.85i)T + (-40.5 + 70.1i)T^{2}
11 1+(−91.4−158.i)T+(−7.32e3+1.26e4i)T2 1 + (-91.4 - 158. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1+206.T+2.85e4T2 1 + 206.T + 2.85e4T^{2}
17 1+(72.5+125.i)T+(−4.17e4+7.23e4i)T2 1 + (72.5 + 125. i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(−127.−73.8i)T+(6.51e4+1.12e5i)T2 1 + (-127. - 73.8i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(−530.−306.i)T+(1.39e5+2.42e5i)T2 1 + (-530. - 306. i)T + (1.39e5 + 2.42e5i)T^{2}
29 1−63.1T+7.07e5T2 1 - 63.1T + 7.07e5T^{2}
31 1+(83.2−48.0i)T+(4.61e5−7.99e5i)T2 1 + (83.2 - 48.0i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(−650.−375.i)T+(9.37e5+1.62e6i)T2 1 + (-650. - 375. i)T + (9.37e5 + 1.62e6i)T^{2}
41 1+1.92e3iT−2.82e6T2 1 + 1.92e3iT - 2.82e6T^{2}
43 1−1.83e3iT−3.41e6T2 1 - 1.83e3iT - 3.41e6T^{2}
47 1+(−1.06e3+1.84e3i)T+(−2.43e6−4.22e6i)T2 1 + (-1.06e3 + 1.84e3i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(3.29e3−1.90e3i)T+(3.94e6−6.83e6i)T2 1 + (3.29e3 - 1.90e3i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(−2.31e3+1.33e3i)T+(6.05e6−1.04e7i)T2 1 + (-2.31e3 + 1.33e3i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(−4.02e3−2.32e3i)T+(6.92e6+1.19e7i)T2 1 + (-4.02e3 - 2.32e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(−1.53e3+886.i)T+(1.00e7−1.74e7i)T2 1 + (-1.53e3 + 886. i)T + (1.00e7 - 1.74e7i)T^{2}
71 1+7.31e3T+2.54e7T2 1 + 7.31e3T + 2.54e7T^{2}
73 1+(4.62e3+8.01e3i)T+(−1.41e7+2.45e7i)T2 1 + (4.62e3 + 8.01e3i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(−4.43e3+7.67e3i)T+(−1.94e7−3.37e7i)T2 1 + (-4.43e3 + 7.67e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1−4.44e3T+4.74e7T2 1 - 4.44e3T + 4.74e7T^{2}
89 1+(120.+69.5i)T+(3.13e7+5.43e7i)T2 1 + (120. + 69.5i)T + (3.13e7 + 5.43e7i)T^{2}
97 1+2.11e3T+8.85e7T2 1 + 2.11e3T + 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

−9.512741555558100139325338660013, −9.165131874865324621324179948378, −7.47845146398820059501032991159, −7.05319313330669140601197768923, −6.31135972603047544555923242152, −5.15443989685876873019391166842, −4.09280431080678688499083341658, −2.89148689966634532534135817190, −1.66655893134059784903955175677, −0.37362184960816110361083052265, 0.851579306327711210762821331526, 2.55819898459292282999754080067, 3.60579897311708397061812239958, 4.58922879166713071643097280978, 5.57698649410427798668113087922, 6.51895944677519678208218303766, 7.34049840028916880375627624078, 8.496243453946845125492421694171, 9.406865554832794034508818400249, 10.02684368612433888157295840066

Graph of the ZZ-function along the critical line