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Properties

Label 2-1400-35.4-c1-0-27
Degree $2$
Conductor $1400$
Sign $0.556 + 0.830i$
Analytic cond. $11.1790$
Root an. cond. $3.34350$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.73 + i)3-s + (−1.73 − 2i)7-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s − 3i·13-s + (−1.73 − i)17-s + (−2.5 − 4.33i)19-s + (−0.999 − 5.19i)21-s + (6.06 − 3.5i)23-s − 4.00i·27-s + 6·29-s + (−2 + 3.46i)31-s + (1.73 − 0.999i)33-s + (4.33 − 2.5i)37-s + (3 − 5.19i)39-s + ⋯
L(s)  = 1  + (0.999 + 0.577i)3-s + (−0.654 − 0.755i)7-s + (0.166 + 0.288i)9-s + (0.150 − 0.261i)11-s − 0.832i·13-s + (−0.420 − 0.242i)17-s + (−0.573 − 0.993i)19-s + (−0.218 − 1.13i)21-s + (1.26 − 0.729i)23-s − 0.769i·27-s + 1.11·29-s + (−0.359 + 0.622i)31-s + (0.301 − 0.174i)33-s + (0.711 − 0.410i)37-s + (0.480 − 0.832i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.556 + 0.830i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.929912874\)
\(L(\frac12)\) \(\approx\) \(1.929912874\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + (-1.73 - i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-6.06 + 3.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.33 + 2.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + (-7.79 + 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.52 + 5.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.46 + 2i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + (10.3 + 6i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.278039013403723284018628345370, −8.800832480979999788638976751633, −8.010946789830557199963913571430, −6.98133612211835829161294311480, −6.35562038286817850766142056728, −4.99809023060385495247102344943, −4.17887144443436498109158338451, −3.21309133647609429473080002735, −2.63836008266889495588638296476, −0.68876421221699789902845641128, 1.62329028416028736711852175576, 2.51435494090858670129710842555, 3.39322805017192450587754902124, 4.48728692404637249036578006919, 5.70032763215459659024571135303, 6.56597762052751732988513436027, 7.28762785501256699315067112705, 8.200065150981936114728115484432, 8.894559422899811342663593439468, 9.371265521355926746773949570991

Graph of the $Z$-function along the critical line