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Properties

Label 2-1400-35.4-c1-0-32
Degree $2$
Conductor $1400$
Sign $-0.994 - 0.107i$
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  + (−1.16 − 0.673i)3-s + (−0.0412 − 2.64i)7-s + (−0.592 − 1.02i)9-s + (−0.705 + 1.22i)11-s − 6.47i·13-s + (2.24 + 1.29i)17-s + (0.581 + 1.00i)19-s + (−1.73 + 3.11i)21-s + (−1.36 + 0.786i)23-s + 5.63i·27-s − 6.22·29-s + (−2.49 + 4.31i)31-s + (1.64 − 0.950i)33-s + (7.34 − 4.23i)37-s + (−4.36 + 7.55i)39-s + ⋯
L(s)  = 1  + (−0.673 − 0.388i)3-s + (−0.0155 − 0.999i)7-s + (−0.197 − 0.342i)9-s + (−0.212 + 0.368i)11-s − 1.79i·13-s + (0.545 + 0.314i)17-s + (0.133 + 0.230i)19-s + (−0.378 + 0.679i)21-s + (−0.284 + 0.164i)23-s + 1.08i·27-s − 1.15·29-s + (−0.447 + 0.775i)31-s + (0.286 − 0.165i)33-s + (1.20 − 0.696i)37-s + (−0.698 + 1.20i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $-0.994 - 0.107i$
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: \((2,\ 1400,\ (\ :1/2),\ -0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5390906620\)
\(L(\frac12)\) \(\approx\) \(0.5390906620\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0412 + 2.64i)T \)
good3 \( 1 + (1.16 + 0.673i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (0.705 - 1.22i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6.47iT - 13T^{2} \)
17 \( 1 + (-2.24 - 1.29i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.581 - 1.00i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.36 - 0.786i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.22T + 29T^{2} \)
31 \( 1 + (2.49 - 4.31i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.34 + 4.23i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + 7.24iT - 43T^{2} \)
47 \( 1 + (9.47 - 5.47i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.0 + 6.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.02 - 1.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.307 - 0.533i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.26 - 3.04i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.27T + 71T^{2} \)
73 \( 1 + (3.04 + 1.75i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.439 - 0.761i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.90iT - 83T^{2} \)
89 \( 1 + (-2.98 - 5.16i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−9.292684271321731863936597677692, −7.990782075640766688626814696235, −7.61630657999556599083731269401, −6.67719544684346102250099446434, −5.78201759942936298041649629104, −5.18436458316356071561052989486, −3.88963231233056590754032269517, −3.07336335661647007339600864297, −1.39177921570861694200418669315, −0.24387707304330304157100125803, 1.82974252095766487049261586765, 2.89855879607238860123546997374, 4.24106487890555426269089305837, 5.00491088672665662345798804705, 5.85715452668463860137371012455, 6.43895498302419907766892484177, 7.62452904173808063070361231287, 8.415892154183163666253427895849, 9.417973122673130304374725509449, 9.722723754912989411968001492901

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