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Properties

Label 2-1400-35.9-c1-0-32
Degree $2$
Conductor $1400$
Sign $0.0989 + 0.995i$
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  + (2.84 − 1.64i)3-s + (−0.0641 + 2.64i)7-s + (3.91 − 6.77i)9-s + (−2.91 − 5.04i)11-s − 2.75i·13-s + (1.73 − i)17-s + (−0.378 + 0.654i)19-s + (4.16 + 7.64i)21-s + (−0.462 − 0.266i)23-s − 15.8i·27-s + 0.823·29-s + (1.28 + 2.23i)31-s + (−16.5 − 9.57i)33-s + (−4.11 − 2.37i)37-s + (−4.53 − 7.85i)39-s + ⋯
L(s)  = 1  + (1.64 − 0.949i)3-s + (−0.0242 + 0.999i)7-s + (1.30 − 2.25i)9-s + (−0.877 − 1.52i)11-s − 0.764i·13-s + (0.420 − 0.242i)17-s + (−0.0867 + 0.150i)19-s + (0.909 + 1.66i)21-s + (−0.0963 − 0.0556i)23-s − 3.05i·27-s + 0.152·29-s + (0.231 + 0.401i)31-s + (−2.88 − 1.66i)33-s + (−0.677 − 0.390i)37-s + (−0.725 − 1.25i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0989 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.0989 + 0.995i$
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: \((2,\ 1400,\ (\ :1/2),\ 0.0989 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.832115399\)
\(L(\frac12)\) \(\approx\) \(2.832115399\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0641 - 2.64i)T \)
good3 \( 1 + (-2.84 + 1.64i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.91 + 5.04i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.75iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.378 - 0.654i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.462 + 0.266i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.823T + 29T^{2} \)
31 \( 1 + (-1.28 - 2.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.11 + 2.37i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 + 0.710iT - 43T^{2} \)
47 \( 1 + (-11.1 - 6.44i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.27 + 4.20i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.70 - 8.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.2 + 5.93i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.04 - 1.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.75 - 8.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.71iT - 83T^{2} \)
89 \( 1 + (-0.878 + 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−8.977023913962124812105936796189, −8.534826408052487263587494610826, −7.956591203371123758128924806561, −7.23575085657940120020100576750, −6.10282117627255973644273373396, −5.43722719253006849499520779344, −3.78698733310990867110421483312, −2.88484303260439451704853712989, −2.45832726820891093215686703134, −0.967125667928411416332934868223, 1.83814770546292863478287242120, 2.70607868897902628676209187000, 3.86872109454489852353407587068, 4.35294417404083084045378412111, 5.18022865617970937184567780194, 6.90266800793245676450606884809, 7.53199186866358788049598331312, 8.133133189167899183510794644576, 9.079723530044556520149336767677, 9.751290824020096789750084606026

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