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Properties

Label 2-700-7.6-c0-0-1
Degree 22
Conductor 700700
Sign ii
Analytic cond. 0.3493450.349345
Root an. cond. 0.5910540.591054
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − i·3-s − i·7-s − 11-s − i·13-s + i·17-s − 21-s − i·27-s + 29-s + i·33-s − 39-s + i·47-s − 49-s + 51-s + 2·71-s + 2i·73-s + ⋯
L(s)  = 1  − i·3-s − i·7-s − 11-s − i·13-s + i·17-s − 21-s − i·27-s + 29-s + i·33-s − 39-s + i·47-s − 49-s + 51-s + 2·71-s + 2i·73-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: ii
Analytic conductor: 0.3493450.349345
Root analytic conductor: 0.5910540.591054
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ700(601,⋅)\chi_{700} (601, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :0), i)(2,\ 700,\ (\ :0),\ i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.91404926370.9140492637
L(12)L(\frac12) ≈\approx 0.91404926370.9140492637
L(1)L(1) 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+iT 1 + iT
good3 1+iT−T2 1 + iT - T^{2}
11 1+T+T2 1 + T + T^{2}
13 1+iT−T2 1 + iT - T^{2}
17 1−iT−T2 1 - iT - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
29 1−T+T2 1 - T + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1−T2 1 - T^{2}
43 1+T2 1 + T^{2}
47 1−iT−T2 1 - iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1+T2 1 + T^{2}
71 1−2T+T2 1 - 2T + T^{2}
73 1−2iT−T2 1 - 2iT - T^{2}
79 1−T+T2 1 - T + T^{2}
83 1−2iT−T2 1 - 2iT - T^{2}
89 1−T2 1 - T^{2}
97 1−iT−T2 1 - iT - T^{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

−10.46066348006947616896795504313, −9.824947976513585554972070817681, −8.254657621720474302069963053615, −7.891889877853086117941671628195, −7.01449192328764480481317833813, −6.18675533791111212741284877628, −5.05183757213150740405063828488, −3.85203172566153858658551399481, −2.54749219910442087433699752843, −1.09331339495056724340748796325, 2.22972192861995293180341740539, 3.36215374059236355005664948042, 4.67109932167493845430517850771, 5.17433253965694296237475362312, 6.35368728470318625647203523095, 7.42766257220495881120390361797, 8.541282495342680240225583349039, 9.270719671494445086390622554280, 9.931782335995926821818685202912, 10.76035020396532464080243620846

Graph of the ZZ-function along the critical line