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Properties

Label 2-700-35.19-c4-0-18
Degree 22
Conductor 700700
Sign −0.964−0.263i-0.964 - 0.263i
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  + (7.14 + 12.3i)3-s + (−32.4 + 36.7i)7-s + (−61.6 + 106. i)9-s + (91.7 + 158. i)11-s + 67.0·13-s + (128. + 223. i)17-s + (447. + 258. i)19-s + (−686. − 139. i)21-s + (775. + 447. i)23-s − 605.·27-s + 1.32e3·29-s + (179. − 103. i)31-s + (−1.31e3 + 2.27e3i)33-s + (−1.75e3 − 1.01e3i)37-s + (479. + 829. i)39-s + ⋯
L(s)  = 1  + (0.794 + 1.37i)3-s + (−0.662 + 0.749i)7-s + (−0.761 + 1.31i)9-s + (0.758 + 1.31i)11-s + 0.396·13-s + (0.446 + 0.773i)17-s + (1.23 + 0.715i)19-s + (−1.55 − 0.315i)21-s + (1.46 + 0.846i)23-s − 0.830·27-s + 1.57·29-s + (0.186 − 0.107i)31-s + (−1.20 + 2.08i)33-s + (−1.28 − 0.740i)37-s + (0.315 + 0.545i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.964−0.263i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.964−0.263i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.964−0.263i-0.964 - 0.263i
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), −0.964−0.263i)(2,\ 700,\ (\ :2),\ -0.964 - 0.263i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 3.0108854473.010885447
L(12)L(\frac12) ≈\approx 3.0108854473.010885447
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+(32.4−36.7i)T 1 + (32.4 - 36.7i)T
good3 1+(−7.14−12.3i)T+(−40.5+70.1i)T2 1 + (-7.14 - 12.3i)T + (-40.5 + 70.1i)T^{2}
11 1+(−91.7−158.i)T+(−7.32e3+1.26e4i)T2 1 + (-91.7 - 158. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1−67.0T+2.85e4T2 1 - 67.0T + 2.85e4T^{2}
17 1+(−128.−223.i)T+(−4.17e4+7.23e4i)T2 1 + (-128. - 223. i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(−447.−258.i)T+(6.51e4+1.12e5i)T2 1 + (-447. - 258. i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(−775.−447.i)T+(1.39e5+2.42e5i)T2 1 + (-775. - 447. i)T + (1.39e5 + 2.42e5i)T^{2}
29 1−1.32e3T+7.07e5T2 1 - 1.32e3T + 7.07e5T^{2}
31 1+(−179.+103.i)T+(4.61e5−7.99e5i)T2 1 + (-179. + 103. i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(1.75e3+1.01e3i)T+(9.37e5+1.62e6i)T2 1 + (1.75e3 + 1.01e3i)T + (9.37e5 + 1.62e6i)T^{2}
41 1+1.97e3iT−2.82e6T2 1 + 1.97e3iT - 2.82e6T^{2}
43 1+1.27e3iT−3.41e6T2 1 + 1.27e3iT - 3.41e6T^{2}
47 1+(1.90e3−3.29e3i)T+(−2.43e6−4.22e6i)T2 1 + (1.90e3 - 3.29e3i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(2.70e3−1.55e3i)T+(3.94e6−6.83e6i)T2 1 + (2.70e3 - 1.55e3i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(−699.+403.i)T+(6.05e6−1.04e7i)T2 1 + (-699. + 403. i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(3.75e3+2.16e3i)T+(6.92e6+1.19e7i)T2 1 + (3.75e3 + 2.16e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(−82.5+47.6i)T+(1.00e7−1.74e7i)T2 1 + (-82.5 + 47.6i)T + (1.00e7 - 1.74e7i)T^{2}
71 1+3.81e3T+2.54e7T2 1 + 3.81e3T + 2.54e7T^{2}
73 1+(332.+576.i)T+(−1.41e7+2.45e7i)T2 1 + (332. + 576. i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(−1.23e3+2.14e3i)T+(−1.94e7−3.37e7i)T2 1 + (-1.23e3 + 2.14e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1−6.31e3T+4.74e7T2 1 - 6.31e3T + 4.74e7T^{2}
89 1+(−5.60e3−3.23e3i)T+(3.13e7+5.43e7i)T2 1 + (-5.60e3 - 3.23e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1−4.76e3T+8.85e7T2 1 - 4.76e3T + 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

−10.03051794592937191816430540370, −9.348783292643145295751491161490, −8.945984906258277382603116197704, −7.86018603815020302339049260113, −6.74493989334044237770734889457, −5.56636325370712558819094868894, −4.67239088939726622111055306626, −3.62361305076583296055541624184, −3.01847662407453733005827428067, −1.59020950309011475268485590058, 0.74389077310869351784058304423, 1.16829711945552189489592530926, 3.01945130112294894142731917304, 3.22153626965043291027569369234, 4.92783034250280850703908012212, 6.46747448365427362327139101594, 6.72369719222831856854745534807, 7.68546951460603752773308030917, 8.563188827774016977238778216846, 9.172305603046846467597109609172

Graph of the ZZ-function along the critical line