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Properties

Label 2-1400-35.4-c1-0-16
Degree $2$
Conductor $1400$
Sign $-0.0771 + 0.997i$
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  + (−1.77 − 1.02i)3-s + (−1.48 + 2.18i)7-s + (0.594 + 1.02i)9-s + (1.78 − 3.08i)11-s + 2.90i·13-s + (3.22 + 1.85i)17-s + (−1.78 − 3.09i)19-s + (4.87 − 2.35i)21-s + (−2.69 + 1.55i)23-s + 3.70i·27-s − 1.57·29-s + (0.382 − 0.661i)31-s + (−6.31 + 3.64i)33-s + (6.06 − 3.50i)37-s + (2.97 − 5.15i)39-s + ⋯
L(s)  = 1  + (−1.02 − 0.590i)3-s + (−0.562 + 0.826i)7-s + (0.198 + 0.343i)9-s + (0.537 − 0.930i)11-s + 0.806i·13-s + (0.781 + 0.450i)17-s + (−0.410 − 0.711i)19-s + (1.06 − 0.513i)21-s + (−0.561 + 0.324i)23-s + 0.713i·27-s − 0.293·29-s + (0.0686 − 0.118i)31-s + (−1.09 + 0.634i)33-s + (0.996 − 0.575i)37-s + (0.476 − 0.825i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0771 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0771 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.0771 + 0.997i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.0771 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7972640970\)
\(L(\frac12)\) \(\approx\) \(0.7972640970\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 2.18i)T \)
good3 \( 1 + (1.77 + 1.02i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1.78 + 3.08i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.90iT - 13T^{2} \)
17 \( 1 + (-3.22 - 1.85i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.78 + 3.09i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.69 - 1.55i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.57T + 29T^{2} \)
31 \( 1 + (-0.382 + 0.661i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.06 + 3.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 + 10.9iT - 43T^{2} \)
47 \( 1 + (-6.38 + 3.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.88 + 1.66i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.07 + 7.04i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.96 - 5.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.1 + 7.00i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + (11.4 + 6.61i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.49 - 2.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + (-2.35 - 4.08i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10.6iT - 97T^{2} \)
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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

−9.161718649210021087854945340116, −8.767596079444032245237796893675, −7.53878766183569402064128415622, −6.70886475417251041759502643766, −5.90064924055309333561109492127, −5.68001895239206183002924184950, −4.26522508563052818764126641032, −3.18772042904826664780948268316, −1.86410456501367984027047503753, −0.44848117973637604752981023764, 1.06579751392095587196935068382, 2.82265889843165068929995765235, 4.08330745025753398210859432769, 4.59219573076877482332553954201, 5.75055014810048870196891273234, 6.25843658134652142078512625797, 7.34938654778656155197354275419, 7.964181003562543092101319398713, 9.296933317989176772445984288266, 10.06669153085982196989028058476

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