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Properties

Label 2-1400-35.9-c1-0-25
Degree $2$
Conductor $1400$
Sign $0.330 + 0.943i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.330 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.330 + 0.943i$
Analytic conductor: \(11.1790\)
Root analytic conductor: \(3.34350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1400,\ (\ :1/2),\ 0.330 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8571657945\)
\(L(\frac12)\) \(\approx\) \(0.8571657945\)
\(L(\frac{3}{2})\) 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

−9.853544576748644376492461665358, −8.347861814874528299313792382454, −7.964790814646868141561078343263, −6.82708818289721519086904105408, −5.96972413819591610267129714264, −5.17153634662819944737670411144, −4.44006574932971259656464994554, −3.58395363244916475216193540501, −1.97601280831452375196762100091, −0.43085976744882913828925056628, 1.25058585158651339437805787567, 2.33219141980804770340483815664, 3.78452168032819384951494684738, 4.96541716030442589235518553448, 5.58670142686039896710079978726, 6.47887112433296232073742489007, 7.01411197030077772484136897056, 8.203476836995063119300897092557, 8.773500362635987673891080849186, 9.727641460871687400619742156702

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