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Properties

Label 2-700-35.19-c4-0-16
Degree 22
Conductor 700700
Sign −0.475−0.879i-0.475 - 0.879i
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  + (3.79 + 6.58i)3-s + (30.9 + 38.0i)7-s + (11.6 − 20.1i)9-s + (100. + 174. i)11-s − 86.3·13-s + (14.2 + 24.6i)17-s + (32.9 + 19.0i)19-s + (−132. + 347. i)21-s + (511. + 295. i)23-s + 792.·27-s − 997.·29-s + (−24.7 + 14.3i)31-s + (−767. + 1.32e3i)33-s + (1.02e3 + 590. i)37-s + (−328. − 568. i)39-s + ⋯
L(s)  = 1  + (0.422 + 0.731i)3-s + (0.630 + 0.775i)7-s + (0.143 − 0.248i)9-s + (0.834 + 1.44i)11-s − 0.511·13-s + (0.0492 + 0.0852i)17-s + (0.0912 + 0.0526i)19-s + (−0.301 + 0.788i)21-s + (0.966 + 0.558i)23-s + 1.08·27-s − 1.18·29-s + (−0.0258 + 0.0148i)31-s + (−0.704 + 1.22i)33-s + (0.746 + 0.431i)37-s + (−0.215 − 0.373i)39-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((−0.475−0.879i) Λ‾(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(5-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+2) L(s)=((−0.475−0.879i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.475 - 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: −0.475−0.879i-0.475 - 0.879i
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), −0.475−0.879i)(2,\ 700,\ (\ :2),\ -0.475 - 0.879i)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.7576402682.757640268
L(12)L(\frac12) ≈\approx 2.7576402682.757640268
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+(−30.9−38.0i)T 1 + (-30.9 - 38.0i)T
good3 1+(−3.79−6.58i)T+(−40.5+70.1i)T2 1 + (-3.79 - 6.58i)T + (-40.5 + 70.1i)T^{2}
11 1+(−100.−174.i)T+(−7.32e3+1.26e4i)T2 1 + (-100. - 174. i)T + (-7.32e3 + 1.26e4i)T^{2}
13 1+86.3T+2.85e4T2 1 + 86.3T + 2.85e4T^{2}
17 1+(−14.2−24.6i)T+(−4.17e4+7.23e4i)T2 1 + (-14.2 - 24.6i)T + (-4.17e4 + 7.23e4i)T^{2}
19 1+(−32.9−19.0i)T+(6.51e4+1.12e5i)T2 1 + (-32.9 - 19.0i)T + (6.51e4 + 1.12e5i)T^{2}
23 1+(−511.−295.i)T+(1.39e5+2.42e5i)T2 1 + (-511. - 295. i)T + (1.39e5 + 2.42e5i)T^{2}
29 1+997.T+7.07e5T2 1 + 997.T + 7.07e5T^{2}
31 1+(24.7−14.3i)T+(4.61e5−7.99e5i)T2 1 + (24.7 - 14.3i)T + (4.61e5 - 7.99e5i)T^{2}
37 1+(−1.02e3−590.i)T+(9.37e5+1.62e6i)T2 1 + (-1.02e3 - 590. i)T + (9.37e5 + 1.62e6i)T^{2}
41 1−571.iT−2.82e6T2 1 - 571. iT - 2.82e6T^{2}
43 1+1.29e3iT−3.41e6T2 1 + 1.29e3iT - 3.41e6T^{2}
47 1+(−975.+1.68e3i)T+(−2.43e6−4.22e6i)T2 1 + (-975. + 1.68e3i)T + (-2.43e6 - 4.22e6i)T^{2}
53 1+(362.−209.i)T+(3.94e6−6.83e6i)T2 1 + (362. - 209. i)T + (3.94e6 - 6.83e6i)T^{2}
59 1+(−1.66e3+961.i)T+(6.05e6−1.04e7i)T2 1 + (-1.66e3 + 961. i)T + (6.05e6 - 1.04e7i)T^{2}
61 1+(−1.88e3−1.09e3i)T+(6.92e6+1.19e7i)T2 1 + (-1.88e3 - 1.09e3i)T + (6.92e6 + 1.19e7i)T^{2}
67 1+(5.44e3−3.14e3i)T+(1.00e7−1.74e7i)T2 1 + (5.44e3 - 3.14e3i)T + (1.00e7 - 1.74e7i)T^{2}
71 1−1.59e3T+2.54e7T2 1 - 1.59e3T + 2.54e7T^{2}
73 1+(1.75e3+3.03e3i)T+(−1.41e7+2.45e7i)T2 1 + (1.75e3 + 3.03e3i)T + (-1.41e7 + 2.45e7i)T^{2}
79 1+(5.19e3−8.99e3i)T+(−1.94e7−3.37e7i)T2 1 + (5.19e3 - 8.99e3i)T + (-1.94e7 - 3.37e7i)T^{2}
83 1−1.12e4T+4.74e7T2 1 - 1.12e4T + 4.74e7T^{2}
89 1+(−6.30e3−3.64e3i)T+(3.13e7+5.43e7i)T2 1 + (-6.30e3 - 3.64e3i)T + (3.13e7 + 5.43e7i)T^{2}
97 1+2.89e3T+8.85e7T2 1 + 2.89e3T + 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

−9.821119508800855304789994116615, −9.406363868714169905794569922514, −8.678011736682664966137740504961, −7.53867071239848208380773376557, −6.74614030247883696958108558720, −5.43382686096420510889026910744, −4.60177571891533713185657889877, −3.76287753246848777318467220216, −2.48806375483300385363383751614, −1.39369672849662514925908610367, 0.64271545147724007825936572053, 1.52469459011882792756359923605, 2.78501900139464078988482431107, 3.93323609478957426599219222041, 4.98910890027246688328813660305, 6.17124010028884561570905092556, 7.14321678168767358320378392996, 7.76330222511176730304613977723, 8.591942523332277927092119567508, 9.407426326371909797625376151659

Graph of the ZZ-function along the critical line